Springback Analysis for Warm Bending of Titanium Tube Based on Coupled Thermal-Mechanical Simulation

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Article Springback Analysis for Warm Bending of Titanium Tube Based on Coupled Thermal-Mechanical Simulation

Guangjun Li 1,*, Zirui He 1, Jun Ma 2 , Heng Yang 2 and Heng Li 2,*

1 Chengdu Aircraft Industry (Group) Co., Ltd., Chengdu 610092, China; [email protected] 2 State Key Laboratory of Solidiﬁcation Processing, School of Materials Science and Engineering, Northwestern Polytechnical University, Xi’an 710072, China; [email protected] (J.M.); [email protected] (H.Y.) * Correspondence: [email protected] (G.L.); [email protected] (H.L.)

Abstract: Titanium bent tubular parts attract extensive applications, thus meeting the ever-growing demands for light weight, high reliability, and long service life, etc. To improve bending limit and forming quality, local-heat-assisted bending has been developed. However, significant springback seriously reduces the dimensional accuracy of the bent tubular parts even under elevated forming temperatures, and coupled thermal-mechanical working conditions make springback behavior more complex and difficult to control in warm bending of titanium tubular materials. In this paper, using warm bending of thin-walled commercial pure titanium tube as a case, a coupled thermal-mechanical finite element model of through-process heating-bending-unloading is constructed and verified, for predicting the springback behavior in warm bending. Based on the model, the time-dependent evolutions of springback angle and residual stress distribution during thermal-mechanical unloading

are studied. In addition, the influences of forming temperature and bending angle on springback angle, thickness variation, and cross-section flattening of bent tubes are clarified. This research

Citation: Li, G.; He, Z.; Ma, J.; Yang, provides a fundamental understanding of the thermal-mechanical-affected springback behavior H.; Li, H. Springback Analysis for upon local-heat-assisted bending for improving the forming accuracy of titanium bent tubular parts Warm Bending of Titanium Tube and structures. Based on Coupled Thermal-Mechanical Simulation. Keywords: titanium tube; warm bending; springback; coupled thermal-mechanical modeling Materials 2021, 14, 5044. https:// doi.org/10.3390/ma14175044

Academic Editor: Mansoo Joun 1. Introduction Titanium bent tubular components are widely used in high-end manufacturing indus- Received: 17 July 2021 tries, such as aircraft, aerospace, nuclear energy, etc., due to their high speciﬁc strength Accepted: 26 August 2021 Published: 3 September 2021 and excellent resistance to corrosion and fatigue [1–3]. With the limited formability of thin-walled titanium tubes under cold bending, a local-heat-assisted bending technology

Publisher’s Note: MDPI stays neutral has been recently applied to achieve the bent tubular parts with a tighter bending radius with regard to jurisdictional claims in and higher forming quality [4–6]. In the tube bending processes, however, one of the published maps and institutional affil- most important problems affecting dimensional accuracy of the formed parts is springback iations. during unloading. It refers to the elastically-driven shape changes that happen when the external loads, such as tools, thermal/electric/magnetic fields, etc., are removed from the deformed part, which directly causes problems such as increased tolerances and variability in the subsequent forming operations, and problems in assembly and in the service of parts [7,8]. Therefore, accurate control of springback is of great importance to the fabri- Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. cation of high-quality parts, which in turn requires a clear understanding of springback This article is an open access article characteristics in the forming process. distributed under the terms and Warm forming has been reported as a possible approach to reduce springback for some conditions of the Creative Commons alloys. However, for titanium and its alloys, the ratio of yield stress to elastic modulus is Attribution (CC BY) license (https:// still at a very high level even at elevated temperatures, resulting in a remarkable springback creativecommons.org/licenses/by/ during warm forming [9,10]. From the view of time series, unloading is the final step in 4.0/). a forming operation so that springback is affected by thermal-mechanical history and

Materials 2021, 14, 5044. https://doi.org/10.3390/ma14175044 https://www.mdpi.com/journal/materials Materials 2021, 14, 5044 2 of 15

presents a pronounced sensitivity to the variables involved in the through-process of both loading (forming) and unloading. In addition, compared with cold forming, the springback behavior in heat-assisted deformation is more complex due to the thermal effect on material properties, contact friction, thermal transferring, etc. [10–12]. All of these issues make springback in warm bending of titanium tubular materials a complicated problem and difficult to analyze and predict. Up to now, there is plenty of research on the springback in tube and sheet forming by using analytical, numerical, and experimental methods. To name a few, Li et al. established an analytical-numerical hybrid modeling framework to clarify the neutral layer shifting phenomena and its underlying mechanisms in tube bending, and pointed out that the neutral layer can be positively controlled by heat-assisted bending [5,13]. Zhan et al. derived an analytical model for springback prediction in titanium tube bending, in which the influences of the initial tube specification and material properties on springback are obtained [14,15]. Li et al. studied the springback behaviors of the high strength titanium tube under multi-die constrained cold rotary draw bending by using the analytical modeling, the explicit/implicit 3D-FE modeling, and experiments, indicating that the angular springback increases linearly with increasing the bending angle, while the radius growth slightly fluctuates with the increase of bending angle at the latter bending stage [16]. Zhang et al. constructed a finite element (FE) model for heating-bending of a large-diameter pure titanium tube [6]. From the view of material property, the influences of anisotropy, the Bauschinger effect, and the degradation effect of elastic modulus on springback of tube bending were also studied [17–20]. For the springback in warm forming of titanium alloys, Ozturk et al. experimentally studied the influence of temperature on springback in bending of a commercial pure titanium (CP-Ti) sheet from room temperature to 300 ◦C, and revealed that springback is dramatically reduced at elevated temperatures [21]. Zong et al. studied springback characteristics in the V-bending process of a Ti-6Al-4V sheet in the hot forming temperature domain, in which the negative springback phenomenon was found at 850 ◦C[22]. Grèze et al. studied the springback behavior in an aluminum alloy at different temperatures by using both experimental and numerical approaches [23]. In this study, a split-ring test is used to calibrate the springback amount and revealed that the effect of temperature tends to reduce the stress gradient in the cup wall that is directly linked to the decrease of the springback opening of the ring. Furthermore, Simões et al. numerically studied the influence of the clearance between the die and the punch on springback using the split-ring test [24]. Saito et al. examined the elastoplastic behavior of 980 MPA nano-precipitation strengthened steel at elevated temperatures ranging from room temperature to 700 ◦C, and further used FE simulation to clarify the effect of stress relaxation and unloading creep on springback in warm V- and U-bending processes [25]. The above studies provide a good knowledge basis for springback analysis. However, tube warm bending-related springback is less reported. Especially for the local-heat- assisted rotary draw bending of thin-walled titanium tubes, both the non-uniform local thermal effect and multi-tool boundaries jointly create a challenge to springback analysis. This paper takes the local-heat-assisted bending of a thin-walled CP-Ti tube as a case, at- tempting to clarify the springback characteristics through the coupled thermal-mechanical modeling approach. Based on the model, the evolutions of residual stress distribution will be revealed, and influences of temperature and bending angle on springback will be identified to provide a fundamental understanding of the springback phenomena in warm bending of titanium tubular materials.

2. Analysis of Local-Heat-Assisted Ti-Tube Bending Processes Local-heat-assisted tube bending is developed by introducing the local thermal ﬁeld to the conventional rotary draw bending process, thus improving the bending formability and forming quality for hard-to-deform tubular materials. Figure1 shows the assembly schematic of warm bending of a large-diameter thin-walled titanium tube. Similarly, the tool system includes pressure die, wiper die, clamp die, bending die, insert die, and Materials 2021, 14, x FOR PEER REVIEW 3 of 16 Materials 2021, 14, 5044 3 of 15

tool system includes pressure die, wiper die, clamp die, bending die, insert die, and man- mandreldrel die diewith with several several flexible ﬂexible balls. balls. The The tube tube is clamped is clamped by bythe the clamp clamp die die and and the the insert insert die dieconnecting connecting to the to the bending bending die. die. By Byrotating rotating the the machine machine arm, arm, the the tube tube can can be be drawn drawn to torotate rotate along along with with the the bending bending die, die, which which make makess the tube the tube bend bend to a certain to a certain curvature curvature struc- structure.ture. At the At same the same time, time, the pressure the pressure die move die movess forward forward to provide to provide a push-assistant a push-assistant force forceto feed to feedmaterial. material. The wiper The wiper die and die andmandrel mandrel die provide die provide support support to avoid to avoid wrinkling wrinkling and andover-flattening over-ﬂattening defects defects duri duringng the thebending bending process. process.

FigureFigure 1.1. SchematicSchematic of of local local heat-assisted heat-assisted bending bending of ofthin-walled thin-walled titanium titanium tube, tube, adapted adapted withwith per- permissionmission from from Springer Springer Nature Nature [6]. [6 ].

InIn the the rotary rotary draw draw bending bending process, process, the the bending bending deformation deformation mainly mainly occurs occurs at at the the tubetube behind behind the the clamping clamping zone, zone, which which is is a a certain certain zone zone of of the the tube tube between between pressure pressure die die andand wiper wiper die. die. Thus,Thus, onlyonly this this zone zone needs needs to to be be heated heated to to the the pre-designed pre-designed temperature temperature distribution.distribution. For For the the clamping clamping zone, zone, there there is no is needno need to heat to heat it since it since it should it should be maintained be main- attained a low at temperature a low temperature to provide to provide enough enough stiffness stiffness for clamping. for clamping. As shown As shown in Figure in1 Figure, the temperature1, the temperature distribution distribution of the tube of the is realized tube is byrealized heat transfer by heat from transfer the heatedfrom the pressure heated diepressure and mandrel die and die. mandrel Electrical die. resistanceElectrical resistance elements thatelements are inserted that are into inserted the holes into arethe usedholes forare heating. used for A heating. series of A thermocouples series of thermocoup are appliedles forare real-timeapplied for monitoring real-time of monitoring temperature of attemperature different positions, at different and positions, a temperature and a temp controlerature system control is developed system is todeveloped automatically to au- adjusttomatically the multi-point adjust the heating multi-point power, heating thus ensuring power, thus the targetedensuring forming the targeted temperature forming field tem- forperature thin-walled field for titanium thin-walled tube bending.titanium tube To avoid bending. the over-heatTo avoid transferthe over-heat to the transfer machine to system,the machine the thermal system, baffle the thermal with ribs baffle and with water-cooling ribs and water-cooling pipes is applied. pipes is Although applied. theAlt- multi-pointhough the multi-point accurate temperature accurate temperature can be sustained can be sustained by the local by the heating local method,heating method, it also leadsit also to leads the non-uniform to the non-uniform temperature temperature distribution distribution of bending of bending tools due tools to thedue transferring to the trans- offerring complex of complex heat. heat. As shown in Figure2, the whole warm bending process can be divided into six As shown in Figure 2, the whole warm bending process can be divided into six stages, stages, viz., assembly and lubricating, local heating, temperature holding, warm bending, viz., assembly and lubricating, local heating, temperature holding, warm bending, tools tools unloading, and natural cooling. The local heating and holding stages ensure the unloading, and natural cooling. The local heating and holding stages ensure the thermal thermal field within the forming temperature window. Then, under the coupled action of field within the forming temperature window. Then, under the coupled action of non- non-uniform temperature field and multi-tool constraints, continuous local elastic-plastic uniform temperature field and multi-tool constraints, continuous local elastic-plastic de- deformation is accumulated for bending the tube to a targeted angle. Finally, the tools formation is accumulated for bending the tube to a targeted angle. Finally, the tools are are removed and the bent tubular part cools down. In cold forming, springback generally removed and the bent tubular part cools down. In cold forming, springback generally only only occurs at the instant of removing tools. However, for warm or hot forming, the non- occurs at the instant of removing tools. However, for warm or hot forming, the non-uni- uniform residual stresses may also create change during the cooling process, which would form residual stresses may also create change during the cooling process, which would induce a dimensional variation. Therefore, the unloading springback in warm bending is a coupledinduce a thermal-mechanical dimensional variation. process. Therefore, Here, wethe nameunloading the springback springback in in warm warm bending bending as is twoa coupled components, thermal-mechanical transient springback, process. andHere, cooling we name springback. the springback In this study,in warm the bending CP-Ti, Gradeas two 3 components, tubes with a nominaltransient dimensionspringback, of and 76.2 cooling mm × springback.1.07 mm (outer In this diameter study, ×thewall CP- thickness) are used as case material in bending.

Materials 2021, 14, x FOR PEER REVIEW 4 of 16

Materials 2021, 14, 5044 4 of 15 Ti, Grade 3 tubes with a nominal dimension of 76.2 mm × 1.07 mm (outer diameter × wall thickness) are used as case material in bending.

FigureFigure 2. 2. Thermal-mechanicalThermal-mechanical solution solution for for local local heat-assisted heat-assisted bending. bending.

3.3. Springback Springback Modeling Modeling for for Tube Tube Warm Warm Bending Bending UnloadingUnloading springback springback is isnormally normally a “final a “ﬁnal step” step” in a informing a forming operation operation so that so it that is it sensitiveis sensitive to the to through-process the through-process history. history. In local-heat-assisted In local-heat-assisted tube bending, tube bending, coupled coupledther- mal-mechanicalthermal-mechanical springback springback is crucially is crucially affected affected by the by heating, the heating, warm warmbending bending as well as as well unloadingas unloading processes. processes. Thus, Thus, from from the theview view of th ofe theFE simulation, FE simulation, it is it necessary is necessary to realize to realize thethe thermal-mechanical thermal-mechanical coupled coupled modeling modeling of ofth thee whole whole above-mentioned above-mentioned process. process. In the In the followingfollowing sections, sections, the the thermal-mechanical thermal-mechanical FE FE model model of ofheating, heating, bending, bending, and and unloading, unloading, basedbased on on the the Abaqus, Abaqus, version version 2016, 2016, will will be pr beesented presented and andthe experimental the experimental details details will be will introduced.be introduced.

3.1.3.1. Coupled Coupled Thermal-Mechanical Thermal-Mechanical Modeling Modeling ConsideringConsidering the the characteristics characteristics of of multi- multi-tooltool constraints constraints and non-uniform temperaturetempera- turefield field inlocal-heat-assisted in local-heat-assisted bending bending of of thin-walled thin-walled titanium titanium tube,tube, a thermal-mechani- thermal-mechanical cal3D-FE 3D-FE model model of heating-bending-unloadingof heating-bending-unloadin isg composed is composed of three of three sub-models, sub-models, viz., heatingviz., heatingand heat and transfer heat transfer model, coupledmodel, coupled thermal-mechanical thermal-mechanical bending bending model and model coupled and thermal-cou- pledmechanical thermal-mechanical springback model. springback The three model. sub-models The three are sub-models sequentially are coupled.sequentially The cou- heating pled.and The heat heating transfer and model heat transfer is based model on the is based implicit on the algorithm implicit andalgorithm considers and considers the full-size thegeometries full-size geometries of tools and of tools tube and blank tube to blan ensurek to theensure simulation the simulation accuracy accuracy of temperature of tem- peraturedistribution. distribution. Due to the Due multi-tool to the multi-tool constraints constraints that induce that complex induce complex contacts incontacts rotary in draw rotarybending, draw the bending, implicit solverthe implicit can hardly solver get can convergence hardly get for convergence solving such for kinds solving of problems. such kindsThus, of the problems. warm bending Thus, the model warm is bending based on model the dynamic is based expliciton the dynamic algorithm, explicit in which algo- the rithm,non-uniform in which temperature the non-uniform is passed temperature from the is passed heating/transfer from the heating/transfer simulation. The simula- tube is tion.discretized The tube by is the discretized coupled temperature-displacement by the coupled temperature-displacement shell elements S4RT shell with elements reduced S4RTintegration with reduced and hourglass integration control. and hourglass The total numbercontrol. The of elements total number is 19,159, of elements in which is the 19,159,element in sizewhich for the the element middle size bending for the zone middle is 2 bending mm × 2.4 zone mm is (longitudinal2 mm × 2.4 mm× (longitu-hoop), and dinalthe element × hoop), sizeand forthe theelement clamping size for zone the clamping and end-straight zone and zone end-straight is 3 mm zone× 2.4 is 3 mm. mm To ×improve 2.4 mm. computationTo improve efficiency,computation the efficiency, rigid body the is usedrigid tobody model is used the tool to model geometry the andtool the geometrysymmetric and semi-tube the symmetric modeling semi-tube method modeling is applied. method The details is applied. such asThe physical details such properties, as physicalmeshing, properties, etc., involved meshing, in the FEetc., modeling involved of in heating the FE and modeling warm bending of heating can beand found warm in [6]. bendingHere, only can thebe found material in [6]. properties Here, only of CP-Tithe material tubular properties material of are CP-Ti introduced. tubular material According areto [introduced.6], the true stress-strainAccording to curves [6], the of true the stress-strain uniform deformation curves of stagethe uniform (before deformation necking) of the stageCP-Ti (before tube are necking) at 25~500 of the◦C, CP-Ti as shown tube are in Figureat 25~5003. It°C, could as shown be found in Figure from 3. this It could figure be that foundthe CP-Ti from present this figure a significant that the CP-Ti improvement present a of significant the elongation improvement and the uniform of the elongation deformation under 150~300 ◦C, which could be possibly used as the temperature window for bending process. By using the Swift hardening equation, σ = K(ε + b)n, strain-stress curves in the uniform deformation stages are fitted to obtain the basic parameters, viz., elastic modulus (E), strength coefficient (K), hardening exponent (n), normal anisotropy exponent (r), and prestrain coefficient (b), as shown in Table1. Materials 2021, 14, x FOR PEER REVIEW 5 of 16

and the uniform deformation under 150~300 °C, which could be possibly used as the temperature window for bending process. By using the Swift hardening equation, 𝜎 Materials 2021, 14, 5044 𝐾𝜀 𝑏 , strain-stress curves in the uniform deformation stages are fitted to obtain5 ofthe 15 basic parameters, viz., elastic modulus (E), strength coefficient (K), hardening exponent (n), normal anisotropy exponent (r), and prestrain coefficient (b), as shown in Table 1.

Figure 3. True stress-strain curves of CP-Ti tube at various temperatures. Figure 3. True stress-strain curves of CP-Ti tube at various temperatures. Table 1.InMechanical consideration properties of the of pronounced CP-Ti tube at normal various temperatures.anisotropy of the CP-3 tube, the Hill48 yield function [26], with the isotropic hardening law fitted by the above described Swift Fundamental Material Parameters Coefficients of Hill’48 Model T (◦C) equation, is used to describe the normally anisotropic deformation in the FE simulation of E (GPa) σ0the(MPa) warm bendingK (MPa) process. Theb Hill yield n function r can be F written Gas: H N 25 82.45 381 650.25 0.0040 0.0970 1.78 0.36 0.36 0.64 3.28 𝑓𝜎 𝐹𝜎 𝜎 𝐺𝜎 𝜎 𝐻𝜎 𝜎 2𝐿𝜎 2𝑀𝜎 2𝑁𝜎 𝜎̄ 0 (1) 100 73.22 326 557.82 0.0035 0.0950 2.00 0.33 0.33 0.67 3.33 150 66.67where 298 subscripts 546.90 1, 2, and 0.00773 refer to the 0.1247 principle 2.30 anisotropic 0.30 axes. 0.30 F, G, H, L, 0.70 M, and 3.39N are 200 66.1the 258 anisotropic 523.21 coefficients, 0.0102 which can 0.1543 be calibrated 2.40 by the 0.29 uniaxial 0.29 tension 0.71 tests at 0°, 3.41 45° 250 64.77 244 513.31 0.0111 0.1658 2.30 0.30 0.30 0.70 3.39 300 61.83and 220 90° with respect 481.15 to the 0.0262 longitudinal 0.2146 direction 2.34 of the 0.30tube. Under 0.30 the plane-stress 0.70 3.40 condition, the Hill’48 model depends only on the four coefficients F, G, H, and N. Various methods could be used to calibrate the coefficients of the Hill’48 plasticity model,In including consideration the yield of the stress-based pronounced method, normal r-value anisotropy based ofmethod, the CP-3 and tube, combined the Hill48 ap- proachyield function using yield [26], stress with and the isotropicr-values [27]. hardening In the present law fitted paper, by the the abover-value described based calibra- Swift tionequation, method is usedwas employed to describe to the identify normally the anisotropiccoefficients. deformation It should be innoted the FEthat simulation tube bend- of ingthe is warm a process bending dominated process. by The axial Hill deformation, yield function as canwell be as written the difficulty as: in experimental testing2 of mechanical2 properties in the2 hoop 2direction2 of the tube,2 the2 in-plan property f (σ) = F(σ22 − σ33) + G(σ33 − σ11) + H(σ11 − σ22) + 2Lσ23 + 2Mσ31 + 2Nσ12 − σ = 0 (1) was reasonably assumed. Consequently, the model parameters in the plane-stress state werewhere finally subscripts determined 1, 2, and by 3 referusing to the the r principle-values obtained anisotropic at different axes. F, Gtemperatures,, H, L, M, and asN ◦ shownare the in anisotropic Table 1. coefficients, which can be calibrated by the uniaxial tension tests at 0 , 45◦ and 90◦ with respect to the longitudinal direction of the tube. Under the plane-stress Tablecondition, 1. Mechanical the Hill’48 properties model of depends CP-Ti tube only at various on the fourtemperatures. coefficients F, G, H, and N. Various methods could be used to calibrate the coefficients of the Hill’48 plasticity Fundamental Material Parameters Coefficients of Hill’48 Model model,T (°C) including the yield stress-based method, r-value based method, and combined 𝒃 approachE (GPa) using 𝝈𝟎 yield(MPa) stress K (MPa) and r-values n [27 ]. In the r presentF paper,G the rH-value basedN calibration25 82.45 method 381 was 650.25 employed 0.0040 to identify0.0970 the1. coefficients.78 0.36 It0.36 should 0.64 be noted3.28 that tube100 bending73.22 is a326 process 557.82 dominated 0.0035 by0.0950 axial deformation, 2.00 0.33 as well0.33 as the0.67 difficulty 3.33 in 150 66.67 298 546.90 0.0077 0.1247 2.30 0.30 0.30 0.70 3.39 experimental testing of mechanical properties in the hoop direction of the tube, the in-plan 200 66.1 258 523.21 0.0102 0.1543 2.40 0.29 0.29 0.71 3.41 property was reasonably assumed. Consequently, the model parameters in the plane-stress 250 64.77 244 513.31 0.0111 0.1658 2.30 0.30 0.30 0.70 3.39 state were finally determined by using the r-values obtained at different temperatures, as 300 61.83 220 481.15 0.0262 0.2146 2.34 0.30 0.30 0.70 3.40 shown in Table1. In the unloading stage, springback occurs when the clamp die is removed from the bent tube. During the process of removing the clamp die, the pressure die is still maintained. After the transient springback is completed, the pressure die is removed. In the transient unloading process, the residual stresses of the bent tube are relieved under a constraint at the end-straight zone. Thus, as shown in Figure4, to simplify the unloading process in FE modeling, only the bent tube is reserved. A fixed constraint is applied at the end- Materials 2021, 14, x FOR PEER REVIEW 6 of 16

In the unloading stage, springback occurs when the clamp die is removed from the bent tube. During the process of removing the clamp die, the pressure die is still main- Materials 2021, 14, 5044 tained. After the transient springback is completed, the pressure die is removed. In 6the of 15 transient unloading process, the residual stresses of the bent tube are relieved under a constraint at the end-straight zone. Thus, as shown in Figure 4, to simplify the unloading process in FE modeling, only the bent tube is reserved. A fixed constraint is applied at the straight zone of the tube. The bent tube part can be imported into the springback model end-straight zone of the tube. The bent tube part can be imported into the springback by “File → Import → Part” from the ODB file of the bending simulation. As the bent model by “File → Import → Part” from the ODB file of the bending simulation. As the tube in the springback model inherits the field information, viz., stress field, strain field, bent tube in the springback model inherits the field information, viz., stress field, strain temperature field, etc., from the bent tube after warm bending deformation, all these state field, temperature field, etc., from the bent tube after warm bending deformation, all these variables should be transferred from the warm bending model to the springback model. state variables should be transferred from the warm bending model to the springback To realize this purpose, the pre-defined field is used to define the field information of model. To realize this purpose, the pre-defined field is used to define the field information the bent tube after bending. Springback is an elastically-driven deformation process so of the bent tube after bending. Springback is an elastically-driven deformation process so that only the elastic properties of materials should be defined. If the material properties that only the elastic properties of materials should be defined. If the material properties are not re-defined, the default elastic properties upon unloading are the same as those are not re-defined, the default elastic properties upon unloading are the same as those in in the deformation stage. However, some studies have reported that the elastic modulus the deformation stage. However, some studies have reported that the elastic modulus of oftitanium titanium upon upon unloading unloading after after a certain a certain level level of plastic of plastic deformation deformation is less isthan less the than mod- the modulusulus in the in theinitial initial loading loading stage stage [28–30]. [28–30 To]. Toimprove improve the the springback springback simulation simulation accuracy, accuracy, thethe modulus modulus degradationdegradation effecteffect cancan be considered by by several several nonlinear nonlinear elastic elastic unloading unloading modelsmodels inin springbackspringback simulationsimulation for improved prediction accuracy accuracy [31–33]. [31–33]. This This work work mainlymainly focuses focuses onon thethe mechanicalmechanical and thermal issues issues related related to to springback springback characteristics characteristics duringduring unloading. unloading. TheThe elasticelastic properties used in in unloading unloading are are considered considered the the same same as as the the constantconstant elastic elastic modulusmodulus inin thethe loadingloading stage.

Heating and local warm-assisted bending processes

Implicit solution for heating and heat transfer

Data Transfer Temperature Explicit solution for thermal- , field transfer Geometry part of bent tube mechanical coupling bending − Field variables: stress, strain, under multi-tool constraints force, displacement, velocity, energy, temperature, etc. Thermal-mechanical coupling unloading process , Material properties if necessary Meshing information Fixed constraint to avoid rigid variation

Implicit solution for thermal-mechanical Springback-induced coupling modeling of angular variation unloading springback

FigureFigure 4. 4.Flowchart Flowchart forfor coupledcoupled thermal-mechanical modeling modeling of of springback. springback.

ToTo simulate simulate the the thermal-mechanical thermal-mechanical springback springback behavior, behavior, the the “Coupled “Coupled Temp-Displace- Temp-Dis- mentplacement (Transient)” (Transient)” analysis analysis step step is employed. is employed. Due Due to to the thesignificant significant springback of of the the titaniumtitanium tube, tube, “Nlgeom”“Nlgeom” isis setset as “On” to accommodate accommodate the the nonlinearity nonlinearity in in displacement displacement andand geometry.geometry. InIn orderorder to control the model st stability,ability, the the dissipated dissipated energy energy fraction fraction is is defaulteddefaulted byby thethe systemsystem andand thethe specificspecific dampin dampingg factor factor of of 0.0002 0.0002 (default (default value) value) is is used used totoguarantee guarantee the convergence. convergence. According According to the to theexperiment, experiment, the time the period time period is set as is 1500. set as 1500. The initial increment size is 0.01 and the maximum increment size is 100, while the minimum increment size is 10−9 and the maximum increment step number is 1. Since only the bent tube part is in the FE model of springback, it is not necessary to consider the contact friction. As mentioned above, only the end-straight zone of the formed tube is fixed by the pressure die during unloading. Thus, a fixed constraint is set to avoid the rigid displacement in springback modeling. As shown in Figure4, the freedom degree of translational direction and rotation direction should be released to ensure free deformation. Materials 2021, 14, x FOR PEER REVIEW 7 of 16

The initial increment size is 0.01 and the maximum increment size is 100, while the minimum increment size is 10−9 and the maximum increment step number is 1. Since only the bent tube part is in the FE model of springback, it is not necessary to consider the contact friction. As mentioned above, only the end-straight zone of the formed tube is fixed by Materials 2021, 14, 5044 the pressure die during unloading. Thus, a fixed constraint is set to avoid the rigid7 ofdis- 15 placement in springback modeling. As shown in Figure 4, the freedom degree of translational direction and rotation direction should be released to ensure free deformation. In addition,In addition, due due to the to thehalf-tube half-tube model model that that is used, is used, the theasymmetric asymmetric boundary boundary is imposed is imposed on theon thetube tube symmetric symmetric surface. surface.

3.2.3.2. Model Model Verification Verification UsingUsing the established thermal-mechanical-coupling FE model, the whole-process simulationsimulation of of heating-bending-unloading heating-bending-unloading for for warm warm rotary rotary draw draw bending bending of of the CP-Ti thin-walledthin-walled tube tube is is carried carried out. out. To To verify the established springback model, the ratio of pseudo strain energy (ALLAE) to internal en energyergy (ALLIE) of the bending model is first first assessedassessed and is far far less less than than 10%, 10%, indicating indicating that the hourglass hourglass effect is very slight and can thusthus be be ignored. ignored. From From the the theoretical theoretical view, view, the coupled model is reliable. Furthermore, Furthermore, typicaltypical bending experimentsexperiments are are conducted conducted at at four four sets sets of formingof forming temperature temperature conditions, condi- ◦ ◦ ◦ ◦ tions,viz., 150 viz.,C, 150 200 °C, C,200 250 °C, 250C, and °C, 300and 300C,to °C, evaluate to evaluate the simulationthe simulation accuracy accuracy of springback. of spring- ◦ back.The bending The bending radius radius is R = is 152.4 R = 152.4 mm ( Rmm= 2(DR), = and2D), the and bending the bending angle angle is 90 is. 90°. FigureFigure 55 illustratesillustrates thethe experimentalexperimental platformplatform for warm bending and the experimen- tallytally formed formed tubes. tubes. It It could could be be found found that that th thee overall trend of springback prediction well meetsmeets the the experiments. However, However, the the prediction prediction error error of the springback angle is still at a highhigh level. level. The The reason reason for for the the error error can be attr attributedibuted to that the constitutive models used inin bending bending and and unloading unloading can can hardly hardly accurately accurately characterize the elastic-plastic behaviors ofof the CP-TiCP-Ti tube. tube. CP-Ti CP-Ti exhibits exhibits a certain a certain extent extent of tension-compression of tension-compression asymmetry, asymmetry, which whichcannot cannot be captured be captured by the by Hill48 the Hill48 yield criterion yield criterion and isotropic and isotropic hardening hardening law used law in used this inwork this [ 34work]. Furthermore, [34]. Furthermore, the unloading the unloadin elasticg moduluselastic modulus presents presents a significant a significant degradation deg- radationand then and trends then to trends be asaturated to be a saturated value with value the with accumulation the accumulation of plastic of strainplastic [ 18strain,28]. [18,28].The former The former can result can inresult the changesin the changes in stress-strain in stress-strain distribution distribution of the of bent the bent tube, tube, and andfurther further affects affects the neutral the neutral layer layer shifting shifting and bendingand bending moment. moment. The latter The latter can directly can directly affect the elastic recovery strain, thus making the springback angle in this study smaller. The affect the elastic recovery strain, thus making the springback angle in this study smaller. above two aspects should be carefully considered in future research for more improved The above two aspects should be carefully considered in future research for more im- springback prediction. proved springback prediction.

(a) (b)

FigureFigure 5. 5. WarmWarm bending bending experiment experiment of of CP-Ti CP-Ti tubes: tubes: (a ()a )experimental experimental platform; platform; (b ()b typical) typical experi- experi- mental bent tubes. mental bent tubes.

InIn this this study, study, to to address address the the simulation simulation er error,ror, a corrected corrected approach is proposed proposed based onon the typical experimentsexperiments and and FE FE simulation. simulation. First, First, an an average average relative relative error error of springbackof spring- backsimulation simulation can be can constructed be constructed as: as: exp pred exp exp pred pred e = ∆θ − ∆θ /∆θ + ∆θ − ∆θ /∆θ /2 (2)

where ∆θpred and ∆θexp are the simulative and experimental springback angles, respectively. Then, a corrected coefﬁcient can be deﬁned as:

α = 1 + e (3) Materials 2021, 14, x FOR PEER REVIEW 8 of 16

exp pred exp exp pred pred e =() Δθθθθθθ −Δ Δ+2 Δ −Δ Δ (2)

Materials 2021, 14, 5044 where Δθ pred and Δθ exp are the simulative and experimental springback angles,8 of 15re- spectively. Then, a corrected coefficient can be defined as: Accordingly, the predicted springback angleα= 1+e can be corrected as: (3)

Accordingly, the predicted springbackpred angle can predbe corrected as: ∆θcorrected = ∆θ · α (4) pred pred Δ=Δ⋅θθα (4) Consequently, from Equation (4), thecorrected prediction of the springback angle can thus be corrected.Consequently, from Equation (4), the prediction of the springback angle can thus be corrected.It can be found from Figure6 that the corrected prediction of springback angles can well agreeIt can withbe found the experimental from Figure 6 resultsthat the with corrected a maximum prediction error of ofspringback less than angles 6.5%. can Li et well al. studiedagree with the the springback experimental rules results of titanium with a tubesmaximu atm different error of bendingless than radius6.5%. Li and et al. bending studied anglesthe springback via experimental, rules of titanium analytical, tubes andat different numerical bending simulation radius and approaches, bending angles indicating via ex- thatperimental, the analytically analytical, and and numerically numerical predictedsimulation springbackapproaches, angle indicating presents that the the same analytically trend withand numerically the experiments predicted even springback though there angle are presents some differencesthe same trend in the with exact the valuesexperiments [16]. Thus,even though a constant there corrected are some coefﬁcient differences is in assumed the exact under values different [16]. Thus, bending a constant angles corrected in the followingcoefficient analysis. is assumed under different bending angles in the following analysis.

5 2.0 Exp. Bending angle: 90° Corrected pred. 4 Pred. Corrected coefficient 1.6

3 1.2

2 0.8 Springback angle [°]

1 0.4 Corrected coefficient [-]

0 0.0 140 160 180 200 220 240 260 280 300 320 Forming temperature [℃] FigureFigure 6. 6.Comparison Comparisonbetween betweenpredicted predicted and and experimental experimental springback springback angles. angles.

4.4. ResultsResults andand DiscussionDiscussion BasedBased onon the the established established FE model,FE model, the thermal-mechanical-affectedthe thermal-mechanical-affected springback springback characteristics,characteristics, including including time-dependent time-dependent evolution evolution of springback of springback angle, angle, stress stress distribution, distribu- influencestion, influences of bending of bending angle onangle springback, on springback, etc., are etc., analyzed are analyzed and discussed and discussed in the follow- in the ingfollowing sections. sections. 4.1. Springback Characteristics during Transient Unloading and Cooling 4.1. Springback Characteristics during Transient Unloading and Cooling 4.1.1. Variation of Springback Angle 4.1.1. Variation of Springback Angle During the thermal-mechanical unloading process, the springback angle significantly increasesDuring first the and thermal-mechanical then trends to a saturated unloading value process, with slight the springback fluctuation. angle Figure significantly7 shows theincreases change first of and springback then trends angle to witha saturated time under value 90with◦ bending slight fluctuation. angle at 250 Figure◦C. 7 As shows the transientthe change springback of springback is completed angle with within time a under very short 90° bending time, the angle logarithm at 250 function °C. As the is used tran- tosient demonstrate springback the is timecompleted coordinate. within a very short time, the logarithm function is used to demonstrateIt can be seenthe time that coordinate. the transient springback is almost completed with 10−3 s, and the springback angle is 2.5099◦. During the remaining time from 10−3 to 1500 s, the springback angle is in a minor fluctuation with a maximum of 0.0544◦, which is about 2.2% of the transient springback. The fluctuation of springback during cooling is quite small, which could be possibly covered by the error of measurement. Even so, it is noted that the dynamic evolution of the stress distribution during the cooling process of the bent tube

might affect the springback angle also. On the whole, the springback is mainly caused by the transient springback, and the cooling springback could be ignored in analysis. Materials 2021, 14, 5044 9 of 15 Materials 2021, 14, x FOR PEER REVIEW 9 of 16

3.5

3.0 2.5099 2.5643

2.5

2.0 Slight fluctuation during thermal-mechanical coupling unloading process 1.5

1.0 Springback angle [°]

0.5 Bending angle: 90° Forming temp.: 250 ℃ 0.0 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 Time [s]

FigureFigure 7.7. SpringbackSpringback vs.vs. unloadingunloading timetime underunder 9090°◦ bending angleangle atat 250250 ◦°C.C.

4.1.2.It Evolution can be seen ofResidual that the transient Stress Distribution springback is almost completed with 10−3 s, and the springbackSpringback angle in is tube 2.5099°. bending During is actually the remaining a process time where from the 10− residual3 to 1500 stresss, the ofspringback the bent tubeangle is is partially in a minor released fluctuation after removing with a themaximum external of tools. 0.0544°, Clarifying which the is about stress variation2.2% of the of thetransient bent tube springback. will be helpful The fluctuation to explore of the springback thermal-mechanical-affected during cooling is quite springback small, which rules. Figurecould 8be shows possibly the covered stress distribution by the error of of the measurement. bent tube at Even different so, it unloading is noted that moments the dy- undernamic 90evolution◦ bending of at the 250 stress◦C. The distribution signiﬁcant du changering the in cooling stress distribution process of occursthe bent within tube onemight second. affect Inthe this springback period, the angle transient also. On springback the whole, is completed,the springback and is the mainly stress caused variation by isthe mainly transient caused springback, by the tool-removal and the cooling induced springback elastic unloading.could be ig Thenored stress in analysis. of the straight zone is dramatically decreased. For the bending zone, the stress of extrados that is close to4.1.2. the Evolution pressure die of isResidual increased, Stress however, Distribution the stress of intrados is reduced. As the natural coolingSpringback goes on, in the tube residual bending stress is actually in the areaa process near where the pressure the residual die in stress the extrados of the bent is continuouslytube is partially increased, released however, after removing there is the no external distinct changetools. Clarifyi in theng stress the ofstress the intrados.variation Thisof the phenomenon bent tube will is causedbe helpful by non-uniform to explore the cooling. thermal-mechanical-affected As the bent tube cools springback down, the strength becomes higher. During the cooling process, however, the deformation of this Materials 2021, 14, x FOR PEER REVIEWrules. Figure 8 shows the stress distribution of the bent tube at different unloading10 mo- of 16 zonements is under constrained 90° bending by other at portions250 °C. The that signific have lowerant change temperatures, in stress thusdistribution leading occurs to an increasewithin one of residual second. stress.In this period, the transient springback is completed, and the stress variation is mainly caused by the tool-removal induced elastic unloading. The stress of the straight zone is dramatically decreased. For the bending zone, the stress of extrados that is close to the pressure die is increased, however, the stress of intrados is reduced. As the natural cooling goes on, the residual stress in the area near the pressure die in the extrados is continuously increased, however, there is no distinct change in the stress of the intrados. This phenomenon is caused by non-uniform cooling. As the bent tube cools down, the strength becomes higher. During the cooling process, however, the deformation of this zone is constrained by other portions that have lower temperatures, thus leading to an increase of residual stress.

FigureFigure 8. 8.Residual Residual stress stress distribution distribution of of bent bent tube tube before before unloading unloading under under 90 90°◦ bending bending at at 250 250◦ C.°C.

4.1.3.4.1.3. Evolution Evolution of of Temperature Temperature Distribution Distribution ForFor the the better better observation observation of of the the change change of of tubetube temperature temperature along along the the axis axisdirection, direction, threethree paths paths includingincluding the extrados path, path, middle middle path, path, and and intrados intrados path path are are chosen chosen as asthe reference of observation. Figure 9 shows the temperature variation of the bent tube along the axis direction when the bending temperature is 250 °C and the bending angle is 90°. The bending deformation zone of the tube is shown in the part between the dotted lines. The temperature of the tube in the axis direction changes greatly in the bending deformation zone near the side of the pressure die. This is because the tube is heated by the pressure die and the mandrel before the warm bending process, but it is directly exposed to the air, along with rapid heat dissipation during the warm bending process. It also can be seen that in 10 s, the temperature variation of the zone that is in contact with the mandrel and pressure die is the greatest. This is due to the highest initial temperature in this zone, which leads to a higher rate of heat dissipation.

Materials 2021, 14, 5044 10 of 15

the reference of observation. Figure9 shows the temperature variation of the bent tube along the axis direction when the bending temperature is 250 ◦C and the bending angle is 90◦. The bending deformation zone of the tube is shown in the part between the dotted lines. The temperature of the tube in the axis direction changes greatly in the bending deformation zone near the side of the pressure die. This is because the tube is heated by the pressure die and the mandrel before the warm bending process, but it is directly exposed to the air, along with rapid heat dissipation during the warm bending process. It also can be seen that in 10 s, the temperature variation of the zone that is in contact with the mandrel Materials 2021, 14, x FOR PEER REVIEWand pressure die is the greatest. This is due to the highest initial temperature in this11 zone, of 16 which leads to a higher rate of heat dissipation.

(b) 250 0 s 1 s 10 s

] 200 100 s

°C 1000 s [ 150

100 Temperature Temperature

0 0 200 400 600 800 1000 Length [mm] (c) (c) 250 0 s 250 1 s 0 s 10 s 1 s 100 s 10 s

] 200 ] 200 1000 s 100 s °C °C [ 1000 s 150 150

100 100 Temperature Temperature [

50 50

0 0 0 200 400 600 800 1000 0 200 400 600 800 1000 Length [mm] Length [mm] Figure 9. Evolution of temperatureFigure 9. Evolution ﬁeld with of temperature unloading process field with under unlo 90ading◦ bending process at under 250 ◦90°C formingbending condition:at 250 °C (a) schematic of temperatureforming paths; condition: (b) temperatures (a) schematic in the extrados;of temperature (c) temperatures paths; (b) temperatures in the middle; in ( dthe) temperatures extrados; (c) tem- in peratures in the middle; (d) temperatures in the intrados. the intrados.

4.2.4.2. Forming Forming Temperature-Related Temperature-Related Springback Characteristics 4.2.1.4.2.1. Effect Effect of of Forming Forming Temperature Temperature on Springback BasedBased on the FE model,model, thethe springbackspringback rulesrules under under different different bending bending angles angles at at differ- dif- ferentent temperatures temperatures are are obtained. obtained. It canIt can be be clearly clearly seen seen from from Figure Figure6 that 6 that when when the the bending bend- ingtemperature temperature is less is less than than 250 ◦250C, the°C, springbackthe springback angle angle increases increases with with the increasethe increase of tem- of temperatureperature and and reaches reaches the maximumthe maximum at 250 at◦ 250C. After °C. After that, thethat, springback the springback angle decreasesangle de- creasesgradually gradually with the with increase the incr ofease temperature of temperature from 250from◦ C250 to °C 300 to ◦300C. Within°C. Within 150~250 150~250◦C, °C,the the springback springback angle angle increases increases at at a slowa slow rate rate with with the the increase increase of of temperature. temperature. As the temperaturetemperature increases, increases, the the deformation deformation resistan resistancece of of the the CP-Ti CP-Ti is is decreased so that the stressstress level level of of the the bent bent tube tube becomes becomes lower, lower, and the material flow ﬂow becomes easier. In this case,case, the the portion portion of of elastic elastic deformation deformation in in the the bending bending deformation deformation zone zone of of the tube may increase,increase, resulting resulting in in the the variation variation of of spring springbackback with with temperature. temperature. When When the the temperature temperature ◦ ◦ rangesranges from from 250 250 °CC to to 300 300 °C,C, the the springback springback angle angle decreases decreases slowly slowly with with the the increase increase of temperature. The is because the temperature of the bending zone is at a higher level, therefore both the yield strength, flow stress, and elastic modulus of the tube decrease significantly, which may lead to the decreases of the elastic recovery.

Materials 2021, 14, 5044 11 of 15

of temperature. The is because the temperature of the bending zone is at a higher level, therefore both the yield strength, ﬂow stress, and elastic modulus of the tube decrease signiﬁcantly, which may lead to the decreases of the elastic recovery.

4.2.2. Effect of Springback on Tube Thickness at Various Forming Temperatures During the bending process, the tangential tension strain, hoop tension strain and normal compression strain of the extrados of the bent tube are induced by the tangential tension stress, hoop tension stress, and normal compression stress, which results in the stretching and thinning in the outer side of the bent tube. The tangential compression, hoop tension, and normal tension strains on the inner side of the tube are induced by the tangential compression stress, hoop compression stress, and normal compression stress, which result in the compression and thickening in the inner side of the bent tube, and even lead to the wrinkles. Due to the recovery of the elastic deformation, both the thinning degree at extrados and the thickening degree at intrados would decrease. As the change of thickness due to springback is elastic deformation, the thickness change is very small. Assuming that the elastic deformation of the formed part is completed unloaded, the springback induced change of thickness could be roughly estimated, varying in the range of about 0.12~0.15%. In addition to the elastic deformation induced thickness change, the thermal expansion could also play a role in the variation of thickness. Along with the cooling process, the thickness of the entire tube will decrease. According to SAE [35], the linear thermal expansion coefﬁcient is 8.6 × 10−6 m/m/◦C within 20~100 ◦C and 9.5 × 10−6 m/m/◦C within 20~100 ◦C. Thus, using the linear thermal expansion coefﬁcient, the thermal expansion induced thickness variation could be roughly estimated as −0.11% within 20~100 ◦C and −0.27% within 20~300 ◦C. Thus, by considering the coupled effect of springback and thermal expansion, the change of thickness at extrados during springback would be smaller than that at intrados. Even so, the maximum change of the thinning/thickening degree of the tube thickness is quite minor—less than about 0.4% based on the above estimation, which is much smaller than the thickness changes due to bending deformation and thus could be neglectable for most cases.

4.2.3. Effect of Springback on Tube Cross-Section at Various Forming Temperatures During tube bending, the tangential tensile stress in the extrados, tangential compres- sive stress in the intrados, and the total circumferential stress are toward the center of the cross-section of the bent tube under the bending moment, thus making the cross-section to be flattened. In this study, we take the bending case with 90◦ to analyze the springback induced effect on the cross-section. Figure 10 shows the change of cross-section flattening along bending direction after springback under 90◦ bending at different temperatures. It can be seen that the maximum cross-section flatting occurs at the bending angle around 10◦ before and after springback. For the four bending temperatures, the flattening degrees first increase and then gradually decrease along the bending direction before springback. After springback, the flatting degree first increases along the bending direction and then reaches a maximum at the position of 10◦, then dramatically decreases and the minimum value occurs at the position of about 60◦. Finally, the flattening degree presents a slow increase and gradually approximates to the flattening degree before springback. Therefore, during the unloading springback process, there is a significant change in the overall distribution of flatting degrees after springback.

4.3. Bending Angle Related Springback Characteristics 4.3.1. Effect of Bending Angle on Springback To explore the effects of bending angle on springback, the change of springback angle at the same forming temperature is identiﬁed and discussed. Using the bending process at 250 ◦C as an example for analysis, the change of springback angle under 45◦, 90◦, and 180◦ could be calculated as 2.75◦, 3.14◦, and 3.21◦, respectively, indicating that the springback angle increases with the bending angle. It is noted that the increase of the springback angle Materials 2021, 14, x FOR PEER REVIEW 12 of 16

which result in the compression and thickening in the inner side of the bent tube, and even lead to the wrinkles. Due to the recovery of the elastic deformation, both the thinning degree at extrados and the thickening degree at intrados would decrease. As the change of thickness due to springback is elastic deformation, the thickness change is very small. Assuming that the elastic deformation of the formed part is completed unloaded, the springback induced change of thickness could be roughly estimated, varying in the range of about 0.12~0.15%. In addition to the elastic deformation induced thickness change, the thermal expansion could also play a role in the variation of thickness. Along with the cooling process, the thickness of the entire tube will decrease. According to SAE [35], the linear thermal expansion coefficient is 8.6 × 10−6 m/m/°C within 20~100 °C and 9.5 × 10−6 m/m/°C within 20~100 °C. Thus, using the linear thermal expansion coefficient, the thermal expansion induced thickness variation could be roughly estimated as −0.11% within 20~100 °C and −0.27% within 20~300 °C. Thus, by considering the coupled effect of springback and thermal expansion, the change of thickness at extrados during springback would be smaller than that at intrados. Even so, the maximum change of the thinning/thickening degree of the tube thickness is quite minor—less than about 0.4% based on the above estimation, which is much smaller than the thickness changes due to bending deformation and thus could be neglectable for most cases.

4.2.3. Effect of Springback on Tube Cross-Section at Various Forming Temperatures During tube bending, the tangential tensile stress in the extrados, tangential com- pressive stress in the intrados, and the total circumferential stress are toward the center of the cross-section of the bent tube under the bending moment, thus making the cross-section to be flattened. In this study, we take the bending case with 90° to analyze the springback induced effect on the cross-section. Figure 10 shows the change of cross-section flat- Materials 2021, 14, 5044 tening along bending direction after springback under 90° bending at different tempera-12 of 15 tures. It can be seen that the maximum cross-section flatting occurs at the bending angle around 10° before and after springback. For the four bending temperatures, the flattening degrees first increase and then gradually decrease along the bending direction before presentsspringback. a nonlinearity After springback, with the the bending flatting angle. degree The first increasing increases ratealong of the springback bending tendsdirec- totion decrease and then at thereaches larger a maximum bending angles. at the position This may of be10°, caused then dramatically by the accelerated decreases stress and relaxationthe minimum at elevated value occurs temperatures. at the position With the of increaseabout 60°. of bendingFinally, the angle, flattening the duration degree time pre- Materials 2021, 14, x FOR PEER REVIEWofsents the wholea slow bendingincrease processand gradually is accordingly approximates increased. to the In flattening this condition, degree the before bent zone13spring- of 16 is

heldback. at Therefore, a high temperature, during the resulting unloading in the springback reduction process, of inner there residual is a stress.significant Consequently, change in thethe increasing overall distribution rate of springback of flatting tends degrees to be after slower springback. with the increase of bending angle.

4.3. Bending3.0 Angle Related Springback Characteristics 4.3.1. Effect of Bending Angle on SpringbackBefore springback 2.5 To explore the effects of bending angle 150 on °C springback, the change of springback angle 2.0 200 °C at the same forming temperature is identifi 250ed °C and discussed. Using the bending process 300 °C at 2501.5 °C as an example for analysis, the change of springback angle under 45°, 90°, and 180°1.0 could be calculated as 2.75°, 3.14°, and 3.21°, respectively, indicating that the springback angle increases with the bending angle. It is noted that the increase of the springback 0.5 angle presentsAfter springback a nonlinearity with the bending angle. The increasing rate of springback tends0.0 to decrease 150 °C at the larger bending angles. This may be caused by the accelerated 200 °C -0.5 250 °C stress (%) degree flattening Cross-section relaxation at elevated temperatures. With the increase of bending angle, the dura- 300 °C tion-1.0 time of the whole bending process is accordingly increased. In this condition, the bent zone is-10 held 0 at 10 a high 20 30 temperature, 40 50 60 resulting 70 80 90 in 100the reduction of inner residual stress. Con- Position from the start bending point (°C) sequently, the increasing rate of springback tends to be slower with the increase of bend- ingFigureFigure angle. 10. 10. Cross-section Cross-section ﬂatting flatting degrees degrees under under different different temperatures. temperatures.

4.3.2.4.3.2. Effect Effect of of Springback Springback on on Thickness Thickness under under Various Various Bending Angles TheThe effects effects of of springback springback on on the the wall wall thickn thicknessess in in the the extrados extrados of of the the bent bent tube tube under under ◦ differentdifferent bending bending angles angles at at 250 250 °CC are are explor exploreded and and discussed. discussed. As As shown shown in in Figure Figure 11a,11a, thethe larger larger the the bending bending angle angle is, is, the the higher higher the the wall wall thinning thinning degree degree in in the the extrados extrados is, is, and and thethe greater greater the the reduction reduction value value of of the the wall wall thinning thinning degree degree in in extrados extrados is is after after springback. springback. Figure 11b shows the evolution of the wall thickening degree in the intrados of the bent Figure 11b shows the evolution of the wall thickening degree in the intrados◦ of the bent tubetube along along the the bending bending direction direction under under differen differentt bending bending angles angles at at 250 250 °C.C. It It can can be be seen thatthat the the wall wall thickening thickening degree degree sharply sharply increases increases when when the the angle angle to to the the initial bending section is less than 25◦, and it is almost unchanged when the angle is greater than 25◦. section is less than 25°, and it is almost unchanged when the angle is greater than 25°. Under the bending angles of 45◦ and 90◦, the wall thickening degree in the intrados of Under the bending angles of 45° and 90°, the wall thickening degree in the intrados of the the bent tube reduces after springback, while for the bending condition at 180◦, the wall bent tube reduces after springback, while for the bending condition at 180°, the wall thick- thickening degree slightly increases. ening degree slightly increases.

(a) 24 45° Before springback (b) 45° After springback 10 20 90° Before springback 90° After springback 180° Before springback 8 16 180° After springback

6 12 45° Before springback 45° After springback 8 4 90° Before springback 90° After springback Wall thinning degree (%) degree thinning Wall 180° Before springback Wall thickening degree(%) 4 2 180° After springback

0 0 0 45 90 135 180 0 40 80 120 160 200 Position from the start bending point (°) Position from the start bending point (°) FigureFigure 11. VariationVariation of of wall wall thickness variation variation under under different different bending bending angles angles at at 250 250 °C:◦C: ( (aa)) thinning thinning of of extrados; extrados; ( (bb)) thickeningthickening of of intrados. intrados.

4.3.3. Effect of Springback on Cross-Section under Various Bending Angles To explore the effects of springback on the cross-section flatting under various bending angles, the simulations at 250 °C are carried out for analysis. Figure 12a shows the change of the cross-section flatting degree along the bending direction before and after springback under different bending angles at 250 °C. It can be seen that the cross-section flatting degrees all decrease after springback under different bending angles. The flatting degrees all first increase and then decrease and finally increase along the bending direction after springback. Cross-section behavior is affected by both bending geometry and

Materials 2021, 14, 5044 13 of 15

4.3.3. Effect of Springback on Cross-Section under Various Bending Angles To explore the effects of springback on the cross-section flatting under various bending angles, the simulations at 250 ◦C are carried out for analysis. Figure 12a shows the change of the cross-section flatting degree along the bending direction before and after springback under different bending angles at 250 ◦C. It can be seen that the cross-section flatting Materials 2021, 14, x FOR PEER REVIEW 14 of 16 degrees all decrease after springback under different bending angles. The flatting degrees all first increase and then decrease and finally increase along the bending direction after springback. Cross-section behavior is affected by both bending geometry and tooling. In tooling.particular, In particular, for the rotary for drawthe rotary bending draw process bending in thisprocess study, in this in addition study, in to addition the clamp to die,the clamppressure die, die, pressure wiper die,die, andwiper bend die, die and applied bend die on theapplied outside on surfacethe outside of the surface tube, thereof the is tube, also theremandrel is also die mandrel with several die with flexible several balls flexible insidethe balls tube. inside The the mandrel tube. The die mandrel could significantly die could significantlyaffect the flattening affect the behavior flattening during behavior bending. during It couldbending. be foundIt could from be thefound three from curves the threeafter springbackcurves after thatspringback there is that a zone there where is a zone the flatteningwhere the degreeflattening is negative. degree is Thisnegative. may Thisattribute may toattribute extension to extension displacement displacement of the mandrel of the rod mandrel with respect rod with to therespect bending to the tangent bend- ingpoint. tangent Thus, point. a local Thus, deformation a local deformation at the end at of the the end mandrel of the mandrel rod could rod be could formed be formed during duringbending, bending, which resultswhich results in the negativein the negative flattening flattening after springback. after springback. It also couldIt also becould found be foundfrom Figure from Figure 12b, that, 12b, the that, maximum the maximum flatting fl degreeatting degree increases increases with bending with bending angle beforeangle beforeand after and springback, after springback, and the and maximum the maximum flatting fla degreetting degree after after springback springback is less is thanless than that thatbefore before springback; springback; the reduction the reduction value value first decreases first decreases and then and increases then increases with the with increase the increaseof the bending of the bending angle. angle.

5 5 45° Before springback 90° Before springback 4 180° Before springback After springback 4 Before springback 3 3 2 2 1

1 Cross-section flatting degree (%) degree flatting Cross-section 0 45° After springback 90° After springback 180° After springback -1 (%) degree flatting cross-section Maximum 0 0 40 80 120 160 200 45 90 180 Position from the start bending point (°) Bending angle (°) (a) (b)

Figure 12. VariationVariation of cross-section flatting ﬂatting unde underr different bending angles at 250 °C:◦C: ( (aa)) distribution distribution along along bending bending angle; angle; (b) maximum ﬂattening.flattening.

5.5. Conclusions Conclusions CombinedCombined withwith representativerepresentative experiments experiments and and coupled coupled thermal-mechanical thermal-mechanical FE mod- FE modeling,eling, the springbackthe springback characteristics characteristics of thin-walled of thin-walled CP-Ti CP-Ti tubes tubes in the in local-heat-assisted the local-heat-as- sistedrotary rotary draw draw bending bending process process were were systematically systematically investigated. investigated. The The main main findings findings are aresummarized summarized as follows: as follows: •• Springback in in local-heat-assisted local-heat-assisted bending bending of ofa CP-Ti a CP-Ti tube tube is a is coupled a coupled thermal-me- thermal- chanicalmechanical process, process, which which can be can divided be divided into transient into transient springback, springback, along with along remov- with ingremoving external external tools and tools cooling-induced and cooling-induced springback springback and natural and natural air cooling. air cooling. By com- By biningcombining the implicit the implicit and andexplicit explicit solving solving algorithms, algorithms, a coupled a coupled thermal-mechanical thermal-mechanical FE modelFE model for forspringback springback prediction prediction is established is established and and verified. verified. •• Based on simulation and experiment, therma thermal-mechanicall-mechanical springback characteristics are revealed: revealed: springback springback mainly mainly occurs occurs upon upon transient transient unloading unloading stage, stage, and andthen thenpre- sentspresents slight slight fluctuation fluctuation with with a maximum a maximum of 2.2% of 2.2% of the of thetotal total springback springback during during air cooling;air cooling; under under the thesame same bending bending angle, angle, springback springback increases increases initially initially and andthen then decreases with the increase of the forming temperature; under the same forming temperature, springback angle increases with a degraded rate at higher bending angles. • Springback induced variations on tube thickness and cross-section flattening are discussed: the thickness variation during springback is attributed to the elastic unloading induced deformation and the cooling induced thermal deformation, which is small and neglectable; the maximum cross-section flattening decreases first and then presents an increasing trend, while it increases with the increase of the bending angle.

Materials 2021, 14, 5044 14 of 15

decreases with the increase of the forming temperature; under the same forming temperature, springback angle increases with a degraded rate at higher bending angles. • Springback induced variations on tube thickness and cross-section flattening are discussed: the thickness variation during springback is attributed to the elastic unloading induced deformation and the cooling induced thermal deformation, which is small and neglectable; the maximum cross-section flattening decreases first and then presents an increasing trend, while it increases with the increase of the bending angle.

Author Contributions: Conceptualization, G.L. and H.L.; methodology, G.L., H.L., Z.H. and J.M.; software, H.Y.; validation, Z.H. and H.Y.; formal analysis, G.L. and Z.H.; investigation, G.L., Z.H., J.M.; writing—original draft preparation, G.L., Z.H., J.M. and H.Y.; writing—review and editing, H.L.; project administration, G.L. and H.L.; funding acquisition, G.L. and H.L. All authors have read and agreed to the published version of the manuscript. Funding: The Commercial Aircraft Research and Development Project of China (MJ-2016-G-64). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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