Springback Prediction of Dieless Forming of AZM120 Sheet Metal Based on Constitutive Model

metals

Article Springback Prediction of Dieless Forming of AZM120 Sheet Metal Based on Constitutive Model

Zijin Wu 1 , Junjie Gong 1,*, Yangdong Chen 2, Jinrong Wang 2, Yuanyuan Wei 1 and Jianhe Gao 1

1 College of Mechanical Engineering, Yangzhou University, Yangzhou 225127, China; [email protected] (Z.W.); [email protected] (Y.W.); [email protected] (J.G.) 2 Jiangsu Yawei Machine-Tool Co., Ltd., Yangzhou 225200, China; [email protected] (Y.C.); [email protected] (J.W.) * Correspondence: [email protected]; Tel.: +86-1366-529-2801

Received: 29 April 2020; Accepted: 8 June 2020; Published: 11 June 2020

Abstract: Springback control is a key issue of the sheet-metal-forming process. In this paper, the mechanism of sheet-metal-forming along the folding trajectory of the computer numerical control (CNC) four-side automatic panel bender was studied, based on the bend-forming springback compensation theory of the power function material model. Firstly, the mechanical property of AZM120 sheet metal standard samples was tested. Then, a theoretical model of springback compensation under plane strain conditions was built, based on the constitutive relationship of the elastic or the elastic-plastic power hardening material. In addition, a sheet-metal-forming trajectory model was designed for sheet metal bending using the vector method. Finally, a laser tracker was used to acquire the folding trajectory, and then the reliability of the trajectory model was verified. On this basis, the influences of geometric and process parameters, such as sheet thickness, forming angle, and bending radius in springback control, were studied according to the theoretical formula and verified by experiments. The proposed method is generally applicable to operation conditions where the bending radius ranges between 1.5 and 6.0 mm and plate thickness ranges from 0.8 to 2.5 mm, and the achieved overall accuracy was more than 89%.

Keywords: springback; bending; analytical model; trajectory control

1. Introduction Sheet-metal-forming nowadays is widely used in the aeronautical, automotive, mechanical and construction industries [1]. With the increasingly fast upgrading of products in the field of manufacturing, the demand is more prominent for product diversification and personalization. The manufacture of sheet metal is developing towards high flexibility and efficiency, and small batch and flexible-surface-forming technology [2]. In traditional die-stamping production, the die design cycle is longer and the manufacturing cost is higher, so it is urgent to develop a technology for forming surface-curved parts with high flexibility and precision in the field of plastic processing. The high-speed bending technology of the automatic panel bender is an efficient and flexible digital forming technology [3–5]. The computer numerical control (CNC) automatic panel bender has the advantages of a short operation path, no overall turnover of workpiece, fast feeding speed, etc. It can realize complete automatic production, from lean manufacturing to mass production. The typical folding types are shown in Figure1, which shows forward bend, negative bend, re-flattening, and large-arc bend, respectively, from left to right.

Metals 2020, 10, 780; doi:10.3390/met10060780 www.mdpi.com/journal/metals Metals 2020, 10, 780 2 of 13 Metals 2020, 10, x FOR PEER REVIEW 2 of 15

Figure 1. Schematic illustration of folding operation. Figure 1. Schematic illustration of folding operation. Springback is a prominent feature of the sheet-metal-forming process, which highly affects the accuracySpringback of metallic is a curvedprominent parts feature [6]. Hill of the [7] firstsheet-metal-forming established a relatively process, accuratewhich highly theoretical affects model the of elastic-plasticaccuracy of metallic bending curved under parts plane [6]. strains. Hill [7] Thisfirst theoreticalestablished modela relatively takes accurate transverse theoretical stress andmodel neutral layerof elastic-plastic movement intobending consideration. under plane Accordingstrains. This to theoretical the Von-Mises model yielding takes transverse criterion stress and nonlinear and kinematicneutral layer hardening movement theory, into Morestinconsideration. et al. Acco [8] re-establishedrding to the Von-Mises the final shapeyielding after criterion springback and by usingnonlinear the curvature kinematic radius hardening after releasingtheory, Morestin residual et stress. al. [8] Zhang re-established et al. [9] developedthe final shape the springback after analyticalspringback model by using of U-shapedthe curvature bent radius parts after according releasing to residual Hill, yielding stress. theZhang criterion et al. [9] and developed plane strain condition.the springback Parsa analytical et al. [10 model] proposed of U-shaped the analytical bent parts calculation according formula to Hill, toyielding predict the the criterion curved and surface springbackplane strain of condition. hyperboloids, Parsa et studied al. [10] the proposed influences the analytical of initial calculation thickness, curvatureformula to radius,predict the tension, andcurved other surface factors springback upon springback, of hype andrboloids, verified studied the accuracy the influences of the springback of initial thickness, prediction curvature formula with radius, tension, and other factors upon springback, and verified the accuracy of the springback experiments and other methods. Using the kinematic hardening material model and Yoshida-Uemori prediction formula with experiments and other methods. Using the kinematic hardening material (YU) model, Hassen et al. [11] put forward a strategy for the reduction of springback based on model and Yoshida-Uemori (YU) model, Hassen et al. [11] put forward a strategy for the reduction variable blankholder force. Jung et al. [12] formulated an elastoplastic material constitutive model to of springback based on variable blankholder force. Jung et al. [12] formulated an elastoplastic accountmaterial for constitutive the evolution model of to the account anisotropy for the of evolution a dual-phase of the steel.anisotropy The combined of a dual-phase isotropic-kinematic steel. The hardeningcombined modelisotropic-kinematic was modified hardening and the influence model was of anisotropy modified and on springbackthe influence prediction of anisotropy accuracy on was analyzed.springback In prediction addition, basedaccuracy on explicitwas analyzed. and implicit In addition, finite element based analysis,on explicit Hu and etal. implicit [13] developed finite a methodelement to analysis, predict theHu springback et al. [13] anddeveloped generate a compensatedmethod to predict forming the trajectories, springback and and Zhang generate et al. [14] proposedcompensated an iterative forming method trajectories, of tool and compensation Zhang et al. based[14] proposed on the displacement an iterative adjustmentmethod of tool method. Licompensation et al. [15] established based on a mathematicalthe displacement model adjus of antment involute method. curve rollerLi et trackal. [15] with established forming clearance a compensation.mathematical Combinedmodel of an with involute the mechanism curve roll ofer stress track relaxation, with forming Jin et al.clearance [16] established compensation. springback andCombined residual with relative the mechanism curvature equations of stress relaxation, of springback. Jin et Theal. [16] springback established and springback residual deflectionand residual models wererelative provided. curvature equations of springback. The springback and residual deflection models were provided.At present, there are several methods to control springback, such as die compensation, pressureAt present, correction, there and are forming several methods process optimization. to control springback, Moreover, such due as todie the compensation, processing characteristics pressure ofcorrection, dieless forming and forming for the pr automaticocess optimization. panel bender, Moreover, higher due technical to the processing requirements characteristics are necessary of for dieless forming for the automatic panel bender, higher technical requirements are necessary for springback control and guarantee of accuracy, so it is of great significance to study the springback of springback control and guarantee of accuracy, so it is of great significance to study the springback of slightly constrained panels free of die flanging. slightly constrained panels free of die flanging. 2. Materials and Methods 2. Materials and Methods 2.1. Material Performance Test of AZM120 2.1. Material Performance Test of AZM120 The mechanical property test is an important method to evaluate metal materials [17]. The AZM120 The mechanical property test is an important method to evaluate metal materials [17]. The super galum, which is coated with a 55% aluminium–zinc alloy on its surface while applying structural AZM120 super galum, which is coated with a 55% aluminium–zinc alloy on its surface while steel as its substrate, is taken as the study object of this paper. By testing the properties of AZM120 applying structural steel as its substrate, is taken as the study object of this paper. By testing the platesproperties in a tensile of AZM120 test and plates nanoindentation in a tensile test test, and the nanoindentation stress-strain curve test, and the the stress-strain load-displacement curve and curve ofthe the load-displacement material were obtained, curve of respectively, the material andwere the obtained, yield strength, respectively, Poisson’s and ratio,the yield hardening strength, index, and elastic modulus of the material were obtained, which provided data for the fitting of the power function curve. Metals 2020, 10, x FOR PEER REVIEW 3 of 15 Metals 2020, 10, x FOR PEER REVIEW 3 of 15 Poisson’s ratio, hardening index, and elastic modulus of the material were obtained, which providedMetalsPoisson’s2020 data, 10 ,forratio, 780 the hardeningfitting of the index, power and function elastic curve. modulus of the material were obtained, which3 of 13 providedIn tensile data test, for the the CMT-4000 fitting of thetype power universal function material curve. testing machine (Shenzhen Silver Fly ElectronicIn Technology tensile test, Co., the Ltd.CMT-4000 Shenzhen, type China)universal was material used to testing stretch machine at a tensile (Shenzhen speed Silverof 10 Fly mm/min,ElectronicIn and tensile theTechnology test,strain the gauge CMT-4000Co., was Ltd. attached Shenzhen, type universalto theChina) tensile materialwas sample. used testing toThe stretch material machine at aproperties tensile (Shenzhen speed of the Silver of 10 Fly samplesElectronicmm/min, were Technology measured.and the strain Figure Co., gauge Ltd. 2 shows Shenzhen, was theattached prepar China) toed the tensile was tensile used samples tosample. stretch and The the at a tensilematerial tensile processes. speedproperties of 10 of mm the/min, andsamples the strain were gauge measured. was attachedFigure 2 shows to the tensilethe prepar sample.ed tensile The samples material and properties the tensile of processes. the samples were measured. Figure2 shows the prepared tensile samples and the tensile processes.

(a) (b) (a) (b) Figure 2. Tensile test: (a)Tensile sample size. (b)The tensile processes. Figure 2. Tensile test: (a) Tensile sample size. (b) The tensile processes. Figure 2. Tensile test: (a)Tensile sample size. (b)The tensile processes. According to the measurement, the plating thickness was only 7 µm, which had very limited According to the measurement, the plating thickness was only 7 µm, which had very limited eﬀect effect on Accordingmaterial properties. to the measurement, Therefore, the the nanoplatingindentation thickness test was was only conducted 7 µm, which on the had substrate. very limited on material properties. Therefore, the nanoindentation test was conducted on the substrate. The TI-950 The TI-950effect ontype material nanoindenter properties. (Hysitron Therefore, Bruker the Co nanorporation,indentation Minneapolis test was MN, conducted U.S.A) was on theadopted, substrate. astype wellThe nanoindenter asTI-950 the typeBerkoviec nanoindenter (Hysitron indenter Bruker (Hysitron (Hysitron Corporation, Bruker Bruker Co Minneapolis rporation,Corporation, Minneapolis MN, Minneapolis USA) MN, was U.S.A)MN, adopted, U.S.A). was as adopted, wellA as the quasi-staticBerkoviecas well method as indenter the wasBerkoviec (Hysitron used for indenter testing. Bruker The(Hysitron Corporation, load reached Bruker Minneapolis the Corporation, maximum MN, value Minneapolis US.A). 5000 mN A quasi-static MN,within U.S.A). 15 s. method A Afterwasquasi-static holding used for the testing.method pressure was The for used load 2 s, forit reached was testing. unloaded the The maximum load within reached 15 value s at the a 5000maximumconstant mN rate. within value Figure 5000 15 s.3 mN Aftershows within holdingthe 15 s. the protrusionpressureAfter holding and for depression 2 s, the it waspressure unloadedof the for indentation 2 s, within it was unloadedand 15 s nanoindentation at a constantwithin 15 rate.s testat a Figureprocesses.constant3 showsrate. Figure the protrusion 3 shows the and depressionprotrusion of and the depression indentation of andthe indentation nanoindentation and nanoindentation test processes. test processes.

(a) (b) (a) (b) Figure 3. Nanoindentation test: (a) Protrusion and depression of the indentation. (b) Nanoindentation Figure 3. Nanoindentation test: (a) Protrusion and depression of the indentation. (b) test processes. NanoindentationFigure 3. Nanoindentationtest processes. test: (a) Protrusion and depression of the indentation. (b) TheNanoindentation load-displacement test processes. curve and the stress-strain curve are shown in Figure4, and the values of The load-displacement curve and the stress-strain curve are shown in Figure 4, and the values the three samples are listed in Table1. of the threeThe samples load-displacement are listed in curveTable 1.and the stress-strain curve are shown in Figure 4, and the values of the three samples are listed in Table 1. Table 1. Mechanical properties of AZM120.

Performance Parameter 1 2 3 Mean Tensile strength R (MPa) 389.6 393.7 393.3 392.2 m Poisson’s ratio (µ) 0.310 0.306 0.336 0.317 Elastic Modulus E (GPa) 199.5 201.5 209.4 203.5 Yield Strength σs (MPa) 320.8 333.3 333.1 329.1 Hardening index n 0.205 0.216 0.208 0.21 Metals 2020,Metals10, 780 2020, 10, x FOR PEER REVIEW 4 of4 of15 13

(a) (b)

FigureFigure 4. Test 4. results: Test results: (a) Load-displacement (a) Load-displacement curve. curve. (b) ( Stress-strainb) Stress-strain curve. curve.

2.2. Springback Compensation TheoryTable for Bending 1. Mechanical Forming properties of AZM120.

2.2.1. Basic AssumptionsPerformance parameter 1 2 3 Mean

ConsideringTensile that thestrength length RmL (MPa)from the pressing 389.6 part to the 393.7 free end 393.3 of the sheet die392.2 is over 5 times as much as the sheet thickness t, namely L 5t, the pure bending formula can be used to calculate the Poisson's ratio (µ) ≥ 0.310 0.306 0.336 0.317 transverseMetals force 2020, 10 bending,, x FOR PEER with REVIEW the results accurate enough to meet the engineering requirements5 of [1815 ,19]. The sheet metalElastic forming Modulus process E can(GPa be）regarded 199.5 as unidirectional 201.5 pure bending,209.4 asshown 203.5 in Figure5. σ Yield Strength s (MPa） 320.8 333.3 333.1 329.1

Hardening index n 0.205 0.216 0.208 0.21

2.2. Springback Compensation Theory for Bending Forming

2.2.1. Basic Assumptions Considering that the length L from the pressing part to the free end of the sheet die is over 5 times as much as the sheet thickness t, namely L ≥ 5t, the pure bending formula can be used to calculate the transverse(a) force bending, with the results accurate enough(b) to meet the engineering requirementsFigureFigure 5. 5.Schematic Schematic [18,19]. diagrams The sheet before before metal and and after afterforming sheet sheet bending:pr bending:ocess ( acan) (Beforea) Beforebe Bending.regarded Bending. ( bas) After (bunidirectional) After bending. bending. pure bending, as shown in Figure 5.

InThe theIn Z unidirectionalthe axis unidirectional shows the pure thickness pure bending bending direction theory, theory, of panel the the following bending.following basicThe basic outer assumptions assumptions surface V are1areV2 madeofmade the plate[20]: [20]: is bent,(1) while Plane(1) Plane the hypothesis: inner hypothesis: surface cross-sections cross-sectionsW1W2 of the at plate at both both is ends compressed.ends of of the the slab slab After performperform bending, relative the N rotation rotation1N2, distance under under h the the actionfromaction ofthe bending neutralof bending moment,axis moment, C1C2 butbecomes but are are still N still1'N perpendicular 2perpendicular'. N is an arbitrary to to the the symmetricpoint symmetric in the planeplane thickness ofof the direction. slab; The tangential(2) Unidirectional(2) engineeringUnidirectional stress strain stress hypothesis: ε hypothesis:of N1N2 supposeunder suppose the no action extrusionno extrusion of bending between between moment the the longitudinal M longitudinal can be expressed layers layers of of the slatsas: the during slats theduring bending the bending moment, moment, there is there no transverse is no transv stress,erse stress, and each and layereach layer in the in slab the slab is in is the in statethe state of unidirectional bending or unidirectional compression. of unidirectional bending or unidirectional compression. ' ' − The Z axis shows the thickness directionN1N2 ofN panel1 N2 bending.h The outer surface V1V2 of the plate is ε = = = h⋅k (1) bent, while the inner surface W W of the plate is compressed. After bending, the N N , distance h from 1 2 N1 N2 R 1 2 the neutral axis C1C2 becomes N1’N2’. N is an arbitrary point in the thickness direction. The tangential where R is the radius of neutral layer curvature, k = 1/R is the neutral layer curvature after bending, engineering strain of N1N2 under the action of bending moment M can be expressed as: and k ≥ 0, h is the distance between a certain layer and the neutral layer.

N10 N20 N1N2 h 2.2.2. Springback Compensation Theoryε = Based− on Power= Function= h kMaterial Model (1) N1N2 R · The stress-strain curve is a tool to study the bending springback problem. In order to simplify wherethe calculationR is the radius and ofanalysis neutral of layer the curvature,complicatedk =stress1/R is-strain the neutral curve layerobtained curvature by the after bending bending, and k 0, h is the distance between a certain layer and the neutral layer. experiment,≥ the curve is usually represented approximately by a function expression. In order to accurately describe the stress-strain relationship, the power function hardening material model is used for derivation, as shown in Figure 6.

Figure 6. Power function material model.

The constitutive relations of power function materials are described as follows E ''εεσ ≤ /E  s σ =  σ (2) σε+−s n' εσ ≥  ssA( ) /E  Ε'

Metals 2020, 10, x FOR PEER REVIEW 5 of 15

(a) (b)

Figure 5. Schematic diagrams before and after sheet bending: (a) Before Bending. (b) After bending.

The Z axis shows the thickness direction of panel bending. The outer surface V1V2 of the plate is bent, while the inner surface W1W2 of the plate is compressed. After bending, the N1N2, distance h from the neutral axis C1C2 becomes N1'N2'. N is an arbitrary point in the thickness direction. The tangential engineering strain ε of N1N2 under the action of bending moment M can be expressed as:

' ' − N1N2 N1 N2 h ε = = = h⋅k (1) N1 N2 R

where R is the radius of neutral layer curvature, k = 1/R is the neutral layer curvature after bending, Metalsand k 2020≥ 0, ,h10 is, 780the distance between a certain layer and the neutral layer. 5 of 13

2.2.2. Springback Compensation Theory Based on Power Function Material Model 2.2.2. Springback Compensation Theory Based on Power Function Material Model The stress-strain curve is a tool to study the bending springback problem. In order to simplify the calculationThe stress-strain and analysis curve is aof tool the to complicated study the bending stress-strain springback curve problem.obtained Inby order the tobending simplify the calculationexperiment, andthe curve analysis is usually of the represente complicatedd approximately stress-strain curveby a function obtained expression. by the bending experiment, the curveIn order is usually to accurately represented describe approximately the stress-str byain a relationship, function expression. the power function hardening materialIn order model to is accurately used for deri describevation, the as stress-strain shown in Figure relationship, 6. the power function hardening material model is used for derivation, as shown in Figure6.

FigureFigure 6.6. Power function function material material model. model.

The constitutive constitutive relations relations of of power power function function materials materials are aredescribed described as follows as follows

('' E εεσ E ε ε σ /E≤ /E = 0 s 0 s σ σ= σs ≤n (2)  σs + A(εσ ) ε σs/E0 (2) σε+−−s En'0 εσ≥ ≥ Metals 2020, 10, x FOR PEER REVIEW  ssA( ) /E 6 of 15  Ε' where n is the hardening index obtained by nanoindentation test, and A is the strengthening coeﬃcient, wheregenerally n is 1–1.15 the hardening for the research index objectobtained AZM120 by nanoindentation plates. test, and A is the strengthening coefficient,For a generally wide plate 1–1.15 (where for widththe researchb is much object larger AZM120 than plates. thickness t), its unidirectional pure bending approximatelyFor a wide plate meets (where the plane width strain b is condition.much larger According than thickness to the t Generalized), its unidirectional Hooke’s pure Law, bending substitute E’ for the elastic modulus E of sheet metal (E = E , where is Poisson’s ratio). approximately meets the plane strain condition.0 1 µAccording2 µ to the Generalized Hooke's Law, − As can be observed from Figure7, when the plate' bendingE is in the elastic range, the formula for substitute E' for the elastic modulus E of sheet metal ( E = , where µ is Poisson's ratio）. the ideal bending moment is 1− μ 2 Z t/2 M = 2 σbhdh (3) 0 where M is the bending moment, σ is the stress, t is the plate thickness, and b is the plate width.

Figure 7. Stress distribution in the thickness direction. Figure 7. Stress distribution in the thickness direction. When the outermost layer of the plate yields, the equation for the elastic stage is σ = εE’=h k E, · · thenAs the can elastic be observed moment fromMe Figureis 7, when the plate bending is in the elastic range, the formula for Z hs the ideal bending moment is 2 Me = 2 E0kbh dh (4) 0 t / 2 M = 2 σbhdh (3) 0 σ where M is the bending moment, is the stress, t is the plate thickness, and b is the plate width. When the outermost layer of the plate yields, the equation for the elastic stage is ' ' σ = εE = h⋅k ⋅ E , then the elastic moment Me is

hs M = 2 Ekbhdh'2 (4) e 0 where k is the curvature of plates, hs is the distance from initial yield layer to the neutral layer.

The cross-section of the bend is plastic deformation when the plate bend ships onto the stage of full plasticity deformation, and the bending moment is Mp, which refers to the plastic limit bending moment. Moreover, springback is caused by both partial plastic deformation and elastic deformation. At this time, the bending moment at the bending part is M. When imposing the bending moment M (Me ≤ M < Mp), layers that are hs away from neutral axis reach the initial yield. The integral bending moment can be given by

ht/2 σ =++−s σσεs n M 2bhdh 2[()]s A bhdh (5) 0 hs E

Metals 2020, 10, 780 6 of 13

where k is the curvature of plates, hs is the distance from initial yield layer to the neutral layer. The cross-section of the bend is plastic deformation when the plate bend ships onto the stage of full plasticity deformation, and the bending moment is Mp, which refers to the plastic limit bending moment. Moreover, springback is caused by both partial plastic deformation and elastic deformation. At thisMetals time, 2020 the, 10, bendingx FOR PEER moment REVIEW at the bending part is M. When imposing the bending moment7 of 15 M (Me M < Mp), layers that are hs away from neutral axis reach the initial yield. The integral bending ≤ moment can be given by σσ3b 2 =−+−−1 σε22ssn t M ssbt'2 2 A()() b h (6) 43Z hs EkZ t/2 E 4 σs n M = 2 σbhdh + 2 [σs + A(ε ) ]bhdh (5) − E M = Φ(k) The relationship between the bending0 momenths and curvature can be expressed as: . When the panel is unloaded after the bending moment M is applied, the unloading process is 1 σ 3b σ n t2 equivalent to the elastic effectM caused= σ bybt2 a reverses bending+ A(ε moments ) b( M', andh 2) M’ = −M. Therefore, the (6) s 2 2 s final curvature after the springback4 is completed− 3E0 k should be− asE follows4 − The relationship between the bending moment and curvature can be expressed as: M = Φ(k).

When the panel is unloaded after the bending' moment-1 MMis applied, the unloading process is equivalent k = Φ（M）- (7) to the elastic eﬀect caused by a reverse bending moment M’' , and M’ = M. Therefore, the ﬁnal curvature E I − after the springback is completed should be as follows where the moment of inertia section I is

1 M k0 = Φ−t / 2(M) (7) 2 I = 2 bh dh− E 0I (8) 0 where the moment of inertia section I is According to the formulas (6)–(8) Z t/2 2 σσ I = 2 (n-1)bh dh22ss (n+ 1) (8) 3 3Aσ (1−++ ) (n 1) 34σσ s0 '' ' =−ss + Etk Etk kk ''323- (9) 2σ − According to the formulasEt (6)–(8) E k t Enn'(1)nn()(1)(2)s ++ Etk' (n+1) 3 (n 1) 2σs 2σs After establishing the curvature3σs springback4σs 3A σfosrmula,− (1 theE tkrelationship) (n + betweenE tk + 1) the curvature k k0 = k + − 0 0 (9) θ − E t E 3k2t3 − 2σ (n 1) and forming angle is analyzed.0 The0 schematic diagramE n( s )of sheet− (n bending+ 1)(n + is2 shown) in Figure 8. 0 E0tk

After establishing the curvature springback formula, the relationship between the curvature k and forming angle is analyzed. The schematic diagram of sheet bending is shown in Figure8.

Figure 8. Bending of sheet metal. Figure 8. Bending of sheet metal. According to the forming method and process, the bending angle and curvature radius of sheet According to the forming method and process, the bending angle and curvature radius of metal are given by sheet metal are given by 360L θ = (10) 3602R°Lπ θ = (10) 2Rπ θ where is the target forming angle, L is the length from the bending moment to the root of plate in the initial bending state, R is the bending radius, and R = 1/K. Substituting Equation (9) into Equation (10), the springback angle can be obtained by

Metals 2020 10 Metals, 2020, 780, 10, x FOR PEER REVIEW 8 of 157 of 13

σσ where θ is the target forming angle, L is the length from(n-1) the bending22ss (n+ 1) moment to the root of plate in the 3 3Aσ (1−++ ) (n 1) ° 34σσ s '' initial bending state,θ ' =−+360R isL the bendingss radius, and R = 1/K. Etk Etk k ''323- (11) Substituting Equationπ (9) into Equation (10), the springback2σ − angle can be obtained by 2 Et E k t '(1)nns ++ Enn()(1)(2)' Etk  (n 1) 2σ (n+1) 2σ  3 − ( s ) (n + s + )  θ 360L 3σs 4σs 3Aσs 1 E tk E tk 1  where ' is theθ springback0 = k angle, and+ R’ = 1/K’ is the springback− 0 bending radius.0  (11)  3 2 3 (n 1)  2π  − E0t E0 k t − n 2σs −  E0 ( E tk ) (n + 1)(n + 2) 2.3. Sheet Forming Trajectory Model 0 where θ is the springback angle, and R’ = 1/K’ is the springback bending radius. This0 section analyzes the edge-folding trajectories of sheet and establishes the trajectory model. 2.3.The Sheetmodel Forming accuracy Trajectory directly Modelaffects the angle of the folding trajectories and the forming quality of sheet metal, and significantly affects the control of springback. ThisIn this section section, analyzes the vector the edge-foldingmethod is used trajectories for modeling. of sheet In order and establishesto ensure the the efficiency trajectory and model. Thefeasibility model of accuracy the model, directly the following affects the assumptions angle of the are folding made: trajectories and the forming quality of sheet metal,(1) and During significantly the forming affects process the control of the of springback.plate, the cross-section thickness of sheet metal is constant;In this section, the vector method is used for modeling. In order to ensure the efficiency and feasibility(2) The of scratch the model, of die the tip following on the surface assumptions of sheet are metal made: is not considered in the sheet forming trajectory.(1) During the forming process of the plate, the cross-section thickness of sheet metal is constant; (2)On The this scratch basis, ofthe die establishmen tip on the surfacet of the of plate sheet folding metal is track not considered model is carried in the sheet out, formingas shown trajectory. in FigureOn 9. this basis, the establishment of the plate folding track model is carried out, as shown in

Figure9.

(a) (b) (c)

Figure 9. Schematic diagram of bending process: (a) Initial state of bending. (b) During bending. Figure 9. Schematic diagram of bending process: (a) Initial state of bending. (b) During bending. (c) (c) After bending. After bending.

As shown in Figure9a, la is the sum of the bend arc length and die-rolling length when the sheet As shown in Figure 9a, la is the sum of the bend arc length and die-rolling length when the is bent to the target angle, lb refers to the length of the straight line in the initial state of bending, and lz referssheet tois thebent total to the length target of theangle, sheet lb bending refers to expansion; the length Figure of the9 bstra is theight schematic line in the diagram initial state established of → basedbending, on theand modellz refers size to vector the total formula. length ofα theis the sheet real-time bending angle expansion; in bending Figure process, 9b is theBA schematicis the die tip α radiusdiagramrm ,established and lg is the based length on ofthe arc model during size bending. vector formula. As shown in is Figurethe real-time9c, , β is angle the target in bending angle, rc is the neutral layer radius, and r is the bending outer radius. The speciﬁc modeling process is as follows process, BA is the die tip radius0 rm , and lg is the length of arc during bending. As shown in Figure 9c, β is the target angle, r is the neutral layer radius, and r is the bending outer , c (rm + ro) 0 rc = i (12) radius. The specific modeling process is as follows 2 · where i refers to the bias coeﬃcient of the neutral()rr layer.+ ri=⋅mo (12) According to Figure9a, the total lengthc of the2 sheet bending expansion is where i refers to the bias coefficient of the neutral layer. lz = lb + la (13) According to Figure 9a, the total length of the sheet bending expansion is where la = rcβ + rmβ. =+ lllzba (13) lz = lb + rcβ + rmβ (14)

l = lz le lg = l + rcβ + rmβ rmα rcα = l + (rc + rm)(β α) (15) f − − b − − b − Metals 2020, 10, 780 8 of 13

where le is the length of die rolling compensation in bending process, and lf is the length of the straight section after the panel is bent. OA→ = OD→ + DT→ + TB→ + BA→ (16)   →  OD = irc   → 3 3  DT = ro(cos( 2 π + α) + i cos( 2 π + α)) = ro(sin α i cos α)  − (17)  →  TB = lf(cos α + i sin α)   → 3 3  BA = rm(cos( π + α) + i cos( π + α)) = rm(sin α i cos α) 2 2 − → ∴ OA = (ro + rm) sin α + l f cos α + i(ro (ro + rm) cos α + l sin α) (18) − f → According to the equation OA = xa + iya, the trajectory coordinates can be expressed as

x = (r + r ) sin α + l cos α a o m f (19) ya = ro (ro + rm) cos α + l sin α − f 3. Results and Discussion In this section, experiments are conducted to verify the accuracy and feasibility of the proposed springback control. Additionally, the parameters aﬀecting the trajectory are analyzed and discussed.

3.1. Verification of Hemming Trajectory and Springback Control Effects Typical processing conditions, i.e., an AZM120 sheet with a width of 1000 mm and a thickness of 1.5 mm, were adopted for verification. Firstly, in order to confirm the consistency between the actual and theoretical edge-folding trajectories, a laser tracker was used to collect the trajectories. The laser tracker works by installing a target ball on the test object and tracking the target ball with the light beam emitted by the tracker, so as to measure and collect the motion trajectory and position relation of the test object. The trajectory parameters selected for the test were as follows: the thickness t was 1.5 mm, the die tip radius rm was 2 mm, the neutral layer radius rc was 2.25 mm, and the bending outerMetals radius 2020, 10r,0 xwas FOR 3PEER mm. REVIEW The specific testing situation is shown in Figure 10. 10 of 15

(a) (b)

Figure 10. Testing situation: (a) Test on site. (b) Target ball Figure 10. Testing situation: (a) Test on site. (b) Target ball A comparison of the result on each trajectory is shown in Figure 11. In order to eliminate the inﬂuenceA comparison of mold of sti theﬀness result on theon each results, trajectory the no-load is shown and full-load in Figure trajectories 11. In order were to collected.eliminate the influence of mold stiffness on the results, the no-load and full-load trajectories were collected.

Figure 11. Trajectory comparison.

In Figure 11, ‘no-load’ and ‘full-load’ show the experimental results of the actual bending trajectories of the four-side automatic panel bender with no load and full load, respectively. The experimental results of ‘no-load’ were used to prove the accuracy of the theoretical trajectory, while those of ‘full-load’ were used to estimate the impact of the actual structure deformation under load on the trajectory. It was found that the folding trajectories with no load were nearly the same as the theoretical ones (as shown in Equation (19), the mold stiffness was not considered). While there is still an 8% bias between those with full load and theoretical ones; this mainly results from the die deformation caused by the working load. Secondly, in order to confirm the forming angle and the corresponding springback angle, the displacement sensor was used to measure the angle of in-place formation, which was compared with the theoretical springback angle, as shown in Figure 12.

Metals 2020, 10, x FOR PEER REVIEW 10 of 15

(a) (b)

Figure 10. Testing situation: (a) Test on site. (b) Target ball

MetalsA comparison2020, 10, 780 of the result on each trajectory is shown in Figure 11. In order to eliminate the9 of 13 influence of mold stiffness on the results, the no-load and full-load trajectories were collected.

Figure 11. Trajectory comparison. Figure 11. Trajectory comparison. In Figure 11, ‘no-load’ and ‘full-load’ show the experimental results of the actual bending trajectoriesIn Figure of11, the ‘no-load’ four-side and automatic ‘full-load’ panel show benderthe experimental with no load results and of full the load,actual respectively. bending trajectoriesThe experimental of the four-side results ofautomatic ‘no-load’ panel were bender used to with prove no the load accuracy and full of load, the theoretical respectively. trajectory, The experimentalwhile those results of ‘full-load’ of ‘no-load’ were usedwere toused estimate to prov thee the impact accuracy of the of actual the theo structureretical deformationtrajectory, while under thoseload of on ‘full-load’ the trajectory. were Itused was to found estimate that the the impact folding of trajectories the actual with structure no load deformation were nearly under the same load as onthe the theoretical trajectory. onesIt was (as found shown that in the Equation folding (19), trajec thetories mold with stiﬀ nessno load was were not considered).nearly the same While as the there theoreticalis still an ones 8% bias (as betweenshown in those Equation with full(19), load the mo andld theoretical stiffness was ones; not this considered). mainly results While from there the is die stilldeformation an 8% bias caused between by those the working with full load. load and theoretical ones; this mainly results from the die deformationSecondly, caused in by order the toworking conﬁrm load. the forming angle and the corresponding springback angle, theSecondly, displacement in order sensor to confirm was used the to forming measure angl the anglee and ofthe in-place corresponding formation, springback which was angle, compared the Metals 2020, 10, x FOR PEER REVIEW 11 of 15 displacementwith the theoretical sensor was springback used to angle, measure as shown the an ingle Figure of in-place 12. formation, which was compared with the theoretical springback angle, as shown in Figure 12.

Figure 12. Sensor schematic in the springback test. Figure 12. Sensor schematic in the springback test. The forming angle of sheet metal is the key to evaluating processing accuracy. Therefore, it is of greatThe significanceforming angle to of study sheet the metal relationship is the key betweento evaluating different processing factors accuracy and the. springbackTherefore, it angle,is of and to greatverify significance the feasibility to study of thethe springbackrelationship controlbetween proposed different infactors this paper.and the springback angle, and to verifyTherefore, the feasibility the eofff ectsthe spring of plateback thickness, control proposed forming angle,in this andpaper. bending radius on springback control areTherefore, further studied. the effects The of bent plate plate thickness, examples form areing shown angle, in and Figure bending 13. radius on springback control are further studied. The bent plate examples are shown in Figure 13.

(a) (b)

Figure 13. Bent plate samples: (a) Sample 1. (b) Sample 2.

3.2. Influence Factors of Springback Control As shown in Figure 14a, AZM120 plates with different thicknesses under typical working conditions (forming angle 90°, bending radius 2.5 mm) were analyzed, comparing the theoretically calculated springback angles with actual ones in practice. Figure 14b shows the forming angles of AZM120 plates with 1.5 mm in thickness and a bending radius of 2.5 mm. In order to make the result comparison more intuitive, the thinning of plates lateral section, the offset of neutral layer, and the hardening behavior of material were not considered [21]. Experimental data are shown in Table 2.

Table 2. Comparison of theoretical and test values under different forming angles and thicknesses.

Bending outer Springback Angle θ ' (°) Thinkness Relative Forming Theoretical Angle θ (°) Radius r0 (mm) t (mm) Experiments Error (%) Model

Metals 2020, 10, x FOR PEER REVIEW 11 of 15

Figure 12. Sensor schematic in the springback test.

The forming angle of sheet metal is the key to evaluating processing accuracy. Therefore, it is of great significance to study the relationship between different factors and the springback angle, and to verify the feasibility of the springback control proposed in this paper. Therefore, the effects of plate thickness, forming angle, and bending radius on springback

controlMetals are2020 further, 10, 780 studied. The bent plate examples are shown in Figure 13. 10 of 13

Metals 2020, 10, x FOR PEER REVIEW 12 of 15

90 3 0.8 1.66 1.76 5.68% 90 3 1 1.33 1.45 8.29% 90 3 1.2 1.10 1.24 11.33% 90 3 1.5 0.88 1.04 15.44% (a) (b) 90 3 2 0.66 0.84 21.50% 90 3Figure Figure 13. Bent 13. Bent plate2.5 plate samples: samples: (a) Sample (0.53a) Sample 1. (b) Sample 1. (b) Sample 2. 0.73 2. 27.35%

3.2. 3.2.Influence30 Influence Factors Factors of Springback of3 Springback Control Control 1.5 0.36 0.44 18.18% 60 3 1.5 0.72 0.82 12.20% As shownAs shown in Figure in Figure 14a, AZM12014a, AZM120 plates plateswith different with di ffthicknesseserent thicknesses under typical under working typical working 90 3 ° 1.5 0.88 1.02 14.08% conditionsconditions (forming (forming angle angle 90 , bending 90◦, bending radius radius2.5 mm) 2.5 were mm) analyzed, were analyzed, comparing comparing the theoretically the theoretically calculatedcalculated120 springback springback angles3 angles with actual with actualones1.5 in ones practice. in practice. 1.21Figure 14b Figure shows 14b the shows 1.30 forming the angles forming 6.92of angles% of AZM120AZM120135 plates plates with with 1.5 mm 1.53 mmin thickness in thickness 1.5and anda bending a bending radius1.37 radius of 2.5 of mm. 2.5 mm. In 1.53order In order to make to make 10.65the the% result resultcomparison comparison more more intuitive, intuitive, the the thinning thinning ofof platesplates laterallateral section, the the offset offset of of neutral neutral layer, layer, and the andhardening the hardening behavior behavior of material of material were were not not considered considered [21 [21].]. Experimental Experimental data data are are shown shown in in Table 2. Table 2.

Table 2. Comparison of theoretical and test values under different forming angles and thicknesses.

Bending outer Springback Angle θ ' (°) Thinkness Relative Forming Theoretical Angle θ (°) Radius r0 (mm) t (mm) Experiments Error (%) Model

(a) (b)

Figure 14. Effects of thickness and forming angle on springback control: (a)Effect of plate thickness. Figure 14. Effects of thickness and forming angle on springback control: (a) Effect of plate thickness. (b)Effect of forming angle. (b) Effect of forming angle. It can be observed that the theoretical and actual springback angles decreased with the increase in It can be observed that the theoretical and actual springback angles decreased with the increase plate thickness, while the gap between the two parameters widened. In addition, the difference between in plate thickness, while the gap between the two parameters widened. In addition, the difference the theoretical and testing results at different springback angles increased gradually. This mainly between the theoretical and testing results at different springback angles increased gradually. This resultedmainly resulted from the from structural the structural deformation deformation and the and diff theerence difference between between the pure the bending pure bending model model established inestablished this study in and this thestudy actual and transversethe actual transverse force bending. force bending. This explanation This explanation is consistent is consistent with the with previous trajectorythe previous verification. trajectory verification. Figure 15a15a shows shows the the influence influence of ofdifferent different bending bending radii radii on the on thespringback springback angle. angle. In order In orderfor for comparisoncomparison and analysis, the the AZM120 AZM120 plates plates of of 1.5 1.5 mm mm thickness thickness and and bending bending at at 90° 90 were◦ were taken taken as as the basicthe basic condition condition to compareto compare the the springback springback angle angle when when the the bendingbending radii were 3, 3, 3.5, 3.5, 4, 4, 4.5, 4.5, 5, 5, 5.5 5.5 and 6and mm. 6 mm. Figure Figure 15b 15b shows shows the the comparison comparison between between the the setpoints setpoints and and actual actual values values of of the the bending bending radius. Experimentalradius. Experimental data are data shown are shown in Table in 3Table. 3.

Table 3. Comparison of theoretical and test values under different bending radius.

Bending outer Forming Thinkness t Relative θ Springback Angle θ ' (°) Angle (°) Radius r0 (mm) (mm) Error (%)

Metals 2020, 10, x FOR PEER REVIEW 13 of 15 Metals 2020, 10, 780 11 of 13 Theoretical Target Test Experiments Table 2. Comparison of theoretical and test values under diModelﬀerent forming angles and thicknesses. 90 3.0 3.84 1.5 0.88 1.02 14.08% Forming Bending outer Thinkness t Springback Angle θ’ (◦) Relative Angle θ ( ) Radius r (mm) (mm) Error (%) 90 ◦ 3.5 0 4.01 1.5 Theoretical 1.02 Model Experiments 1.12 8.69% 90 904.0 34.42 0.81.5 1.66 1.17 1.76 1.28 5.68%8.67% 90 904.5 34.98 11.5 1.33 1.32 1.45 1.39 8.29%5.37% 90 3 1.2 1.10 1.24 11.33% 90 5.0 5.61 1.5 1.46 1.56 6.29% 90 3 1.5 0.88 1.04 15.44% 90 905.5 35.91 21.5 0.66 1.61 0.84 1.7 21.50%5.39% 90 906.0 36.35 2.51.5 0.53 1.75 0.73 1.81 27.35%3.05% 30 3 1.5 0.36 0.44 18.18% 60 3 1.5 0.72 0.82 12.20% 90 3 1.5 0.88 1.02 14.08% 120 3 1.5 1.21 1.30 6.92% 135 3 1.5 1.37 1.53 10.65%

(a) (b)

Figure 15. Effect of the bending radius on springback control: (a)Effect of bending outer radius. Figure 15. Effect of the bending radius on springback control: (a) Effect of bending outer radius. (b) (b) Setpoints and actual values of bending outer radius. Setpoints and actual values of bending outer radius. Table 3. Comparison of theoretical and test values under different bending radius. The study of the bending radius indicated that the trend of theoretical and experimental results is almostForming consistent.Bending With outer the Radius increaset0 (mm) in theThinkness bending radius,Springback springback Angle increased,θ’ (◦) and theRelative gap Angle θ ( ) t (mm) Error (%) between theoretical◦ Target and tested data Testdecreased. As shown in Theoretical Figure 15b, Model the consistency Experiments between the data of90 the theoretical 3.0 and testing 3.84 bending radi 1.5us improved with 0.88 the enlarging 1.02 of the bending 14.08% radius.90 The main reason 3.5 for this was 4.01 that the bending 1.5 moment increased 1.02 with the 1.12 radius increases. 8.69% 90 4.0 4.42 1.5 1.17 1.28 8.67% In this case, the bending force required reduced, which facilitated the reduction of overall rigidity 90 4.5 4.98 1.5 1.32 1.39 5.37% deformation90 of the automatic 5.0 panel 5.61bender. 1.5 1.46 1.56 6.29% 90 5.5 5.91 1.5 1.61 1.7 5.39% 4. Conclusions90 6.0 6.35 1.5 1.75 1.81 3.05% This study presented an analytical model, in which the relationship between the power functionThe study material of themodel bending and the radius springback indicated was that taken the into trend consideration, of theoretical and and the experimental dieless forming results istrajectory almost consistent. was studied. With The theform increaseing radius in and the the bending in-position radius, angle springback after being increased, rebounded and can the be gap betweenaccurately theoretical predicted and by testedthis model. data decreased.Meanwhile, Asthe shown influences in Figure of sheet 15b, thickness, the consistency bending between radius, the dataandof forming the theoretical angle in and springback testing bendingcontrol were radius studied improved by comparing with the enlarging the theoretical of the calculations bending radius. Thewith main the reasonexperimental for this tests. was The that following the bending conclusion moments can increased be derived with based the radius on the increases. obtained results: In this case, the bending(1) The force uniaxial required tension reduced, tests were which carried facilitated out to theobtain reduction the true of stress overall–strain rigidity relationship deformation of of thematerial, automatic which panel can bender.develop the elastic-power hardening model more accurately and can be used to

Metals 2020, 10, 780 12 of 13

4. Conclusions This study presented an analytical model, in which the relationship between the power function material model and the springback was taken into consideration, and the dieless forming trajectory was studied. The forming radius and the in-position angle after being rebounded can be accurately predicted by this model. Meanwhile, the influences of sheet thickness, bending radius, and forming angle in springback control were studied by comparing the theoretical calculations with the experimental tests. The following conclusions can be derived based on the obtained results: (1) The uniaxial tension tests were carried out to obtain the true stress–strain relationship of material, which can develop the elastic-power hardening model more accurately and can be used to establish the constitutive relationship of AZM120 materials. Furthermore, the results of nanoindentation tests were carried out to replenish material performance parameters; (2) An established springback compensation trajectory model was tested on an automatic panel bender by a laser tracker and displacement sensors. For different bending parameter combinations, the model had good generalization performance, and the precision was controlled within 0.2 . ± ◦ The theory was highly consistent with testing, which provided basic data for springback compensation and prediction; (3) The results of the experiments showed that the average absolute error between the theoretical and actual springback angle was less than 11%. Therefore, the adopted constitutive models can achieve an allowable relative error, if geometric and process parameters are appropriately selected; (4) According to the analysis results of the various variables and the test of bending trajectory, it was found that with the thickness increases and the bending radius decreases, the bending moment increases, which causes mechnical deformation and concessions, having a significant impact on the springback angle compensation trajectory. The research on the compensation of mechanical deformation on springback is also a significant direction for the future.

Author Contributions: Conceptualization, Z.W. and J.G. (Junjie Gong); Methodology, Y.C.; Resources, J.W. and Y.W.; Writing–original draft, Z.W.; Writing–review & editing, J.G. (Junjie Gong) and J.G. (Jianhe Gao). All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

1. Liewald, M.; Hönle, S.; Sindel, M. Surface roughening of an aluminum 6016 alloy during bending and hemming. Int. J. Mater. Form. 2016, 9, 203–213. [CrossRef] 2. Cui, X.; Xiao, A.; Du, Z.; Yan, Z.; Yu, H. Springback reduction of l-shaped part using magnetic pulse forming. Metals 2020, 10, 390. [CrossRef] 3. Jia, B.; Wang, W.-W. Shape accuracy analysis of multi-point forming process for sheet metal under normal full constrained conditions. Int. J. Mater. Form. 2018, 11, 491–501. [CrossRef] 4. Kim, H.-K.; Kim, W.-J. A springback prediction model for warm forming of aluminum alloy sheets using tangential stresses on a cross-section of sheet. Metals 2018, 8, 257. [CrossRef] 5. Hill, K. Smart manufacturing drives ﬁve key changes. Manuf. Eng. 2019, 163, 14. 6. Prates, P.A.; Adaixo, A.S.; Oliveira, M.C. Numerical study on the eﬀect of mechanical properties variability in sheet metal forming processes. Int. J. Adv. Manuf. Tech. 2015, 96, 561–580. [CrossRef] 7. Hill, R. The Mathematical Theory of Plasticity; Oxford university press: London, UK, 1950. 8. Morestin, F.; Boivin, M.; Silva, C. Elasto plastic formulation using a kinematic hardening model for springback analysis in sheet metal forming. J. Mater. Proc. Tech. 1996, 56, 619–630. [CrossRef] 9. Zhang, D.J.; Cui, Z.S.; Ruan, X.Y.; Li, Y.Q. An analytical model for predicting springback and side wall curl of sheet after U-bending. Comput. Mater. Sci. 2007, 38, 707–715. [CrossRef] 10. Parsa, M.H.; al ahkami, S.N.; Pishbin, H.; Kazemi, M. Investigating spring back phenomena in double curved sheet metals forming. Mater. Des. 2012, 41, 326–337. [CrossRef] Metals 2020, 10, 780 13 of 13

11. ul Hassan, H.; Maqbool, F.; Güner, A.; Hartmaier, A.; Khalifa, N.B.; Tekkaya, A.E. Springback prediction and reduction in deep drawing under influence of unloading modulus degradation. Int. J. Mater. Form. 2016, 9, 19–633. [CrossRef] 12. Jung, J.; Jun, S.; Lee, H.-S.; Kim, B.-M.; Lee, M.-G.; Kim, J.H. Anisotropic hardening behaviour and springback of advanced high-strength steels. Metals 2017, 7, 480. [CrossRef] 13. Zhu, H.; Chang, X.; Jung, D.W. The generation of the forming path with the springback compensation in the CNC incremental forming. Int. J. Mater. Form. 2018, 11, 455–470. [CrossRef] 14. Zhang, Z.; Wu, J.; Zhang, S.; Wang, M.; Guo, R.; Guo, S. A new iterative method for springback control based on theory analysis and displacement adjustment. Int. J. Mech. Sci. 2016, 105, 330–339. [CrossRef] 15. Li, Z.; Shu, X. Involute curve roller trace design and optimization in multipass conventional spinning based on the forming clearance compensation. J. Manuf. Sci. Eng. 2019, 141, 091007. [CrossRef] 16. Jin, L.; Yang, Y.-F.; Li, R.-Z.; Cui, Y.W.; Jamil, M.; Li, L. Study on springback straightening after bending of the U-section of TC4 material under high-temperature conditions. Materials 2020, 13, 1895. [CrossRef][PubMed] 17. Yue, Z.; Qi, J.; Zhao, X.; Badreddine, H.; Gao, J.; Chu, X. Springback prediction of aluminum alloy sheet under changing loading paths with consideration of the influence of kinematic hardening and ductile damage. Metals 2018, 8, 950. [CrossRef] 18. Ruderman, M.S. Fluid Dynamics and Linear Elasticity. In A First Course in Continuum Mechanics; Springer International Publishing: Cham, Switzerland, 2019. [CrossRef] 19. Xue, X.; Liao, J.; Vincze, G.; Pereira, A.B.; Barlat, F. Experimental assessment of nonlinear elastic behaviour of dual-phase steels and application to springback prediction. Int. J. Mech. Sci. 2016, 117, 1–15. [CrossRef] 20. Zaczynska, M.; Kolakowski, Z. The influence of the internal forces of the buckling modes on the load-carrying capacity of composite medium-length beams under bending. Materials 2020, 13, 455. [CrossRef][PubMed] 21. Nader, A. Springback and fracture in v-die air bending of thick stainless steel sheets. Mater. Des. 2000, 21, 217–236.

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