metals
Article Investigation of the Geometry of Metal Tube Walls after Necking in Uniaxial Tension
Chong Li, Daxin E *, Jingwen Zhang and Ning Yi School of Materials Science and Engineering, Beijing Institute of Technology, Beijing 100081, China; [email protected] (C.L.); [email protected] (J.Z.); [email protected] (N.Y.) * Correspondence: [email protected]; Tel.: +86-136-6137-4667
Academic Editor: Enrique Louis Received: 10 February 2017; Accepted: 14 March 2017; Published: 17 March 2017
Abstract: In order to characterize the deformation and true stress–strain relation of metal tubes, the geometry of tube walls after necking in uniaxial tension need to be determined. The paper investigated the necking process of metal tube. A large number of tensile tests and finite element analysis of 1Cr18Ni9Ti tubes with different sizes were conducted. It was found that the geometry of outer tube wall in the necking region can be described using a logistic regression model. The final geometry of the tube is determined by original tube diameter and wall thickness. The offset of tube walls are affected by two competing factors: volume constancy and necking. The offset distances of outer and inner walls are mainly affected by original wall thickness. The length of the necking zone is more influenced by original tube diameter. Tube elongation at fracture increases slightly as tube diameter gets larger, while the wall thickness has almost no impact on the elongation.
Keywords: tube tension; necking; surface profile; tube diameter; wall thickness
1. Introduction Metal tubes have been widely used in many industrial applications because of their special characteristics including light weight, high strength, energy absorption [1], etc. There are increasing numbers of tubular parts in automobiles, ships and aircrafts that replace solid parts. Also in oil and gas fields, forming of pipelines is an extremely important issue. In the design stage of tube application, the prediction of tube forming failure is a key task. Finite element method is an indispensable tool to avoid many trial and error iterations [2]. The stress–strain relationship of the material is one of the most important input parameters to ensure the accuracy of finite element calculation, especially when plastic deformation occurs [3–5]. Due to the hollowed structure, the mechanical properties of metal tubes are different from the sheet metal [6] or round metal bars [7]. Tao and Yang described the tube sample’s mechanical properties using the measurement of dog bone specimens which are cut from the tube samples [8]. However, the specimens after cut off may exhibit different mechanical behavior since the axial symmetry has been destroyed. Another assumption is that metal tubes share the same mechanical property as metal bars [9]. Apparently, these approximations are not accurate when analyzing the forming process of metal tubes [10]. Only the mechanical properties obtained from direct measurement from the tube specimens can characterize the forming mechanics of tube metals with a higher accuracy [11–15]. Rathnaweera et al. [16] studied the performance of aluminum tubes and Terocore® hybrid structures in quasi-static three-point bending. They observed that the bottom surface failure (tensile) from structures is one of the failure modes during bending and it occurred when the bottom surface of the outer tube reached the ultimate tensile strain. Finite element analysis (FEA) was carried out in a subsequent study by Hanssen et al. [17] using LS-DYNA (Livermore Software Technology Corporation, Livermore, CA, USA). However, a correlation between experimental results was not obtained, since
Metals 2017, 7, 100; doi:10.3390/met7030100 www.mdpi.com/journal/metals Metals 2017, 7, 100 2 of 14 they were unable to use a failure criterion in their model. They also pointed out that predicting failure by using damage mechanics on non-filled metal tubes is a complicated task in itself. Shi et al. [18] simulated the necking and fracture of tube under hydro forming. In the modeling approach, they determined the localized necking strain by examining the deformed configuration during tension and assumed the hoop strain was uniformly distributed in the circumferential direction. They claimed that the current model could not differentiate the deformations’ development in the circumferential direction from the axial direction. Kofiani [19] proved that the necking simulation for tubes is very sensitive to the introduced material behavior and the element size. In order to get more accurate simulated results for various tube forming processes, the more accurate stress–strain curve is the main difficulty at this stage. Moreover, the research focus on the most basic deformation of metal tubes, unaxial tension, is still far from complete. A standard uniaxial tensile test can achieve the true stress–strain curves up to the uniform elongation point. Once necking occurs, the conventional measurement methods such as using strain gauge are no longer valid due to the stress triaxiality [20]. Thus, using tensile test data directly as input into FEA will inevitably lead to prediction results bias. Numerous studies were conducted to characterize the material’s behavior after necking [20–30]. Wecould get some inspiration and reference from those of metal sheets and round bars. Wierzbicki et al. [22–26] developed a complete necking and fracture predictive technology for steel sheets. The inverse methods after necking for the simulation purposes were proposed. Bridgman’s method is one of the widely accepted approaches, which generates analytical stress–strain relations by considering the geometry of the neck, such as curvature of the profile and diameter [27]. This method was further developed by other researches and has been applied to tensile specimens with both circular cross sections and rectangular cross sections [28–30]. Ling [28] calculated the true stress–strain functions from engineering stress–stain data of tensile strips by utilizing weighted averages of upper and lower bound of stress–strain curves. However, few literatures have reported applying this method in tubes. Apparently, the post-necking stress–strain relationships of tubes are crucial when designing the tube forming process such as hydro forming [31] or bending [32]. One of the difficulties when implementing Bridgman’s method is the accurate measurement of necking profile. If a simple formula could be developed to describe the curvature profile of the neck, the derivation of the analytical model would extremely convenient. In this study, an analytical formula was proposed to depict the curvature profile, which could easily be used in combination with Bridgman’s equation in the future to derive the post-necking stress–strain relationships of tubes. The geometry of the neck—curvature profile and diameter—during uniaxial tension of tubes was investigated. Experiments and FEA were conducted to investigate the geometrical evolution of the neck and the smallest diameter of the neck just before or at fracture. Experiments, FEA and analytical formula exhibit excellent agreement on the curvature profile depiction. In addition, the neck diameter at fracture and elongation of different tube dimensions in uniaxial tension was also discussed, which is a good industrial reference for design analysis.
2. Materials and Methods
2.1. Material and Test Device A conventional uniaxial tensile test was conducted first to obtain the stress–strain behavior of the tubes. Instead of cutting sections from tube wall, full cross-section tubes were tested directly to ensure the results can reflect the actual mechanical response of the specimen. A commonly used material 1Cr18N9Ti (C ≤ 0.12%, Si ≤ 1.0%, Mn ≤ 2.0%, P ≤ 0.035%, S ≤ 0.03%, Ti 0.50%–0.80%, Ni 8.00%–11.00%, Cr 17.00%–19.00%) was selected in this study. The tube size is defined by its outer diameter (OD) and inner diameter (ID) or wall thickness (WT). The tube sizes used in tensile test are listed in Table1. At least 3 specimens were tested for each size. Metals 2017, 7, 100 3 of 14 Metals 2017, 7, 100 3 of 14
Table 1.1. SizesSizes ofof 1Cr18Ni9Ti1Cr18Ni9Ti tubetube samplessamples selectedselected inin experiment.experiment.
OD0 WT0 6 mm 8 mm 10 mm 12 mm 14 mm OD0 WT0 6 mm 8 mm 10 mm 12 mm 14 mm 0.6mm ○ × × × × 0.6mm × × × × 1 mm ○ ○ ○ ○ ○ 1 mm # 1.5 mm 1.5 mm##### ○ ○ × ×× ×× × 2 mm 2 mm## ○ ○ × ×× ×× × 2.5 mm 2.5 mm## ○ ×× × ×× ×× × # WTWT00denotes denotes the the original original tube tube wall wall thickness. thickness. OD0 denotes OD0 denotes the original the original outer tube outer diameter. tube diameter. “ ” means “ the○” tubemeans in this dimension is selected in experiment while “×” means not. the tube in this dimension is selected in experiment while “×” means not. #
TubeTube specimens specimens were were cut cut based based on on the the Standard Standard GB/T GB/T 228-2002 228-2002 [[3333].]. In order order to to ensure ensure the the uniaxialuniaxial tensiletensile condition,condition, twotwo metalmetal cylindricalcylindrical plugsplugs werewere insertedinserted fromfrom thethe twotwo openopen endsends ofof eacheach tube.tube. TheThe plugs plugs were were specially specially machined machined with with a round a round top top for easyfor easy operation. operation. During During tension, tension, the grips the clampedgrips clamped the two the ends two of ends the tubeof the with tube plugs with inserted.plugs inserted. The dimension The dimension of tube of samples tube samples and the and plugs the areplugs shown are shown in Figure in Figure1a. Previous 1a. Previous test experience test experience shows shows that fracturethat fracture may may occasionally occasionally not not occur occur in thein the middle middle of the of the tube. tube. In order In order to ensure to ensure that thethat fracture the fracture occurs occurs within within the gauge the gauge length, length, two more two pairsmore ofpairs marked of marked lines (Gauge lines (Gauge II and III) II and were III) drawn were on drawn the surface on the of surface each specimen of each specimen besides the besides usual onethe usual in the one middle in the part middle (Gauge part I). (Gauge Once the I). fractureOnce the occurs fracture outside occurs Gauge outside I, GaugeGauge III, orGauge III could II or beIII selectedcould be for selected gauge length for gauge measurement length measuremen according totthe according fracture location. to the fracture WDW-E100 location. universal WDW testing-E100 machineuniversal (Jinan testing Shijin machine Group (Ji Co.nan Ltd.,Shijin Jinan, Group China) Co. Ltd. was, Jinan used, for China the) test, was Figure used for1b. the Constant test, Figure loading 1b. speedConstantv = loading 2.5 mm/min speed was v = selected.2.5 mm/min Tension was selected. stopped Tension once tube stopped samples once fractured. tube samples Strain fractured. level was recordedStrain level with was an recorded extensometer with untilan extensometer 5% extension. until 5% extension.
FigureFigure 1. 1Specimens. Specimens and and test test device: device: ((aa)) dimensiondimension ofof thethe tubetube sample;sample; ( b)) universal testing testing machine. machine.
2.2.2.2. MechanicalMechanical PropertiesProperties FigureFigure2 2shows shows the the mechanical mechanical properties properties of tubeof tube 1Cr18Ni9Ti 1Cr18Ni9Ti obtained obtained from from tensile tensile test. test. From From the engineeringthe engineering stress–strain stress–strain curves curves in Figurein Figure2a, 2 ita, can it can be be seen seen that that tube tube samples samples with with different different sizes sizes exhibitexhibit very similar similar mechanical mechanical response. response. For For simplification, simplification, we we picked picked the thestress stress–strain–strain curve curve of OD of0 = 8 mm and WT0 = 1 to represent the mechanical response of all the tube specimens and to be used in OD0 = 8 mm and WT0 = 1 to represent the mechanical response of all the tube specimens and to befinite used element in finite simulation element simulationas input. The as input.effect of The the effect deviation of the of deviation stress–strain of stress–strain curves on the curves necking on thewill necking be studied will in be future. studied Figure in future. 2b shows Figure true2b shows stress true–strain stress–strain relation before relation necking before as necking calculated as σ = 퐾ε푛 calculated(solid line) (solid and the line) extrapolated and the extrapolated relation after relation necking after (dotted necking line). (dotted The line). power The law power equation law equation σwas= Kusedεn was to fit used the tomaterial’s fit the material’s stress–strain stress–strain curve, where curve, K is where the strengthK is the coefficient strength coefficient and n is the and strainn is thehardening strain hardening index. The index. mechanical The mechanical properties properties coefficients coefficients are list in areFigure list in2b. Figure 2b.
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Figure 2.2. MaterialMaterial properties: ( a) Engineering stress stress–strain–strain curves curves of of different different sizes sizes 1Cr18Ni9Ti 1Cr18Ni9Ti tubes tubes;; Figure 2. Material properties: (a) Engineering stress–strain curves of different sizes 1Cr18Ni9Ti tubes; (b) True stress–strainstress–strain curvecurve andand mechanicalmechanical propertiesproperties ofof 1Cr18Ni9Ti1Cr18Ni9Ti withwith ODOD0 == 8 mm WT0 = 1. 1. (b) True stress–strain curve and mechanical properties of 1Cr18Ni9Ti with OD0 0= 8 mm WT0 =0 1.
2.3.2.3. Deformation Deformation Deformation Pattern Pattern Pattern of of the the Tube Tube Walls Walls AfterAfter tensile tensile test, test, all all the the samples samples were were cut along thethe axisaxis of of the the tube. tube. Figure Figure 3 33shows showsshows the thethe shapes shapesshapes of ofthe the the tube tube tube walls wallswalls at at fracture fracture fracture area area area with with with different different different outer outer diametersdiameters diameters and and wall wall and thicknesses. thicknesses. wall thicknesses. It Itcan can be be clearly It clearly can be seenclearlyseen that that seen both both that inner inner both diameter innerdiameter diameter and and outer outer and diameterdiameter outer diameter at neckingnecking at necking region region shrank regionshrank after shrank after tube tube after fracture. fracture. tube fracture.Here Here weHerewe define define we definethe the original original the original inner inner and and inner outer outer and diametersdiameters outer diameters are IDID00 andand are OD OD ID0 0and andand the ODthe inner 0innerand and and the outer innerouter diameters diameters and outer f f 0 afterdiametersafter fracture fracture after are are fracture ID ID andf and are OD OD ID,f ,respectively. andrespectively. ODf, respectively. FigureFigure 33aa Figureshowsshows 3 thatthata shows the the shrinkage shrinkage that the shrinkageratio ratio of ofOD OD ratio ((OD ((OD of0 OD– – f 0 0 OD((ODOD)/0ODf)/−ODOD) 0increases) fincreases)/OD0) asincreases as the the original original as the wallwall original thicknessthickness wall thickness WTWT0 increases.increases. WT0 increases.The The dimensional dimensional The dimensional change change of ofID change ID is is slightlyofslightly ID is slightlydifferent. different. different. F romFrom the the From observation observation the observation ofof FigureFigure of 3 Figurea,,b,b, itit 3 cancana,b, be be it canfound found be that foundthat the the that shrinkage shrinkage the shrinkage ratio ratio of ofID ratio ID ((ID0 0 – IDf f)/ID0 0) decreases as the thickness-diameter ratio WT0/OD0 0 0increases. More details are ((IDof ID – ((ID ID0)/−IDID) f)/ID decreases0) decreases as the as thickness the thickness-diameter-diameter ratio ratio WT WT/OD0/OD increases.0 increases. More More details details are discussedarediscussed discussed later. later. later.
FigureFigure 3. 3Geometries. Geometries of of tube tube wallswalls at fracture area area with with (a (a) )different different wall wall thicknesses, thicknesses, and and (b) ( bdifferent) different outer diameters. Figureouter diameters. 3. Geometries of tube walls at fracture area with (a) different wall thicknesses, and (b) different outer diameters.
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2.4.2.4. Finite Finite Element Element Model Model CommercialCommercial finite finite element element code code Abaqus Abaqus 6. 6.1010 (SIMULIA (SIMULIA by by Dassault Dassault Systèmes, Systèmes, Waltham, Waltham, MA, MA, USAUSA)) was was used used to to model model the the uniaxial uniaxial tension tension of of tubes. tubes. Only Only a a quarter quarter tube tube with with 12 12 mm mm length length was was modeledmodeled to to save save computation computation time, time, Fi Figuregure 44.. Three-dimensionalThree-dimensional hexahedral solid elements C3D8R werewere used used in in the the model. model. Two Two-dimensional-dimensional axial axial symmetric symmetric elements elements were were not not used used in in this this study study becausebecause the the model model will will be be used used in in future future bending bending simulations, simulations, which which are are not not axial axial symmetric symmetric forming forming proceprocesses.sses. In order order to to accurately accurately capture capture the stress–strain the stress–strain state during state during necking necking and fracture, and fracture,a consistent a consistentfiner element finer size element (0.02mm size× (0.020.02 mm mm ×× 0.020.02 mm mm ×in 0.02 necking mm in zone) necking was zone) used towas mesh used the to critical mesh areathe criticallocally area for alllocally the tube for all sizes. the tube The material’ssizes. The material’s stress–strain stress curve–strain and curve material and properties material properties in Figure 2inb Figurewas used 2b was as FEA used input. as FEA Although input. Although anisotropy anisotropy can be caused can be by caused the rolling by the process rolling of process the tube, of the it is tube,difficult it isto difficult quantify to thequantify anisotropy the anisotropy coefficient. coefficient. Thus, it is Thus, assumed it is inassumed FEA modeling in FEA thatmodeling the tube that is theisotropic. tube is All isotropic. six degrees All ofsix freedom degrees were of freedom fixed at were the bottom fixed surface.at the bottom A 2.5 mm/min surface. A velocity 2.5 mm/min was set velocityas the boundary was set condition as the boundary at the top condition surface, which at the is top the surface, same as thewhich loading is the condition same as in the experiment. loading conditionSymmetric in boundaryexperiment. conditions Symmetric were boundary applied conditions to enforce were the axial applied symmetry. to enforce the axial symmetry.
FigureFigure 4 4.. FiniteFinite element element model model of of tube tube tension tension..
In order to trigger the necking process, a defect was put into the model manually at the middle In order to trigger the necking process, a defect was put into the model manually at the middle of the tube (mesh refinement area), where OD0 is 0.05 mm smaller than the nominal tube diameter. of the tube (mesh refinement area), where OD is 0.05 mm smaller than the nominal tube diameter. The simulation will stop once fracture is initiated0 so that the evolution of the tube’s geometry during The simulation will stop once fracture is initiated so that the evolution of the tube’s geometry during the necking process can be tracked. Thus, we still need to be able to identify the fracture initiation the necking process can be tracked. Thus, we still need to be able to identify the fracture initiation point point even though fracture formation is not the focus of our study. In the FEA model, damage even though fracture formation is not the focus of our study. In the FEA model, damage initiation, initiation, damage evolution and element removal embedded in ABAQUS solver were introduced to damage evolution and element removal embedded in ABAQUS solver were introduced to signal the signal the ending point of the simulation. ending point of the simulation. Ductile criterion for damage initiation was utilized, as shown in Equation 1 [34]. Ductile criterion for damage initiation was utilized, as shown in Equation 1 [34]. dε̅푝푙 ωD = ∫ pl = 1 (1) Z ε̅푝푙(dηε, ε̅̇푝푙) ωD = D = 1 (1) pl . pl ε (η, ε ) where ωD is a state variable that increases monotonicallyD with plastic deformation. The equivalent ε̅푝푙 η ε̅̇푝푙 ε̅푝푙 plastic strainω at the onset of damage, , was a function of stress triaxiality and strain rate . where D is a state variable that increases푝푙 monotonically with plastic deformation. The equivalent was equal to the strain hardening index ε̅ = n = 0.39 [35], which was calculated from the material’s. pl εpl η ε εpl stressplastic–strain strain curve. at the The onset stress of damage, triaxiality for, was tensile a function test was of 0.33. stress The triaxiality damage evolutionand strain was rate defined. pl was equal to the strain hardening푝푙 index ε = n = 0.39 [35], which was calculated from the material’s based on plastic displacement, 푢̅D , which was defined with the evolution equation: stress–strain curve. The stress triaxiality for tensile test was 0.33. The damage evolution was defined pl 푢̅̇ 푝푙 = 퐿ε̅̇푝푙 (2) based on plastic displacement uD , which was defined with the evolution equation: where L is the characteristic length of the element. Assuming a linear evolution of the damage . pl . pl u = Lε pl variable with effective plastic displacement and the effective plastic displacement at failure, u̅f (2)= 0.035 was calibrated using the test data of tube with OD0 = 8 mm and WT0 = 1. Then the damage variable increases according to Equation 3. The maximum degradation was set.
Metals 2017, 7, 100 6 of 14 where L is the characteristic length of the element. Assuming a linear evolution of the damage variable pl with effective plastic displacement and the effective plastic displacement at failure, uf = 0.035 was calibrated using the test data of tube with OD0 = 8 mm and WT0 = 1. Then the damage variable increases according to Equation (3). The maximum degradation was set.
. pl . pl . Lε u Metals 2017, 7, 100 d = = 6 of 14 pl pl (3) u f u f 퐿ε̅̇푝푙 푢̅̇ 푝푙 pl 푑̇ = = To ensure u = 0.035 was applicable in tubes푝푙 with푝푙 different sizes, the element sizes were(3) kept f 푢̅푓 푢̅푓 uniform in all FEA models. 푝푙 To ensure 푢̅푓 = 0.035 was applicable in tubes with different sizes, the element sizes were kept 3. Resultsuniform and in Discussionall FEA models.
In3. Results order to and compare Discussion the results with ease, we have to pick a suitable coordinate system. In this study, the origin O of the coordinate is selected at the center point of the straight line which coincides In order to compare the results with ease, we have to pick a suitable coordinate system. In this with the tube wall of the straight tube section. X = 0 corresponds to the minimum OD location after study, the origin O of the coordinate is selected at the center point of the straight line whichf coincides necking and fracture. Positive y direction points opposite to the tube axis. Due to the fact that the tube with the tube wall of the straight tube section. X = 0 corresponds to the minimum ODf location after deformationnecking and is almost fracture. symmetric Positive y direction about the pointsy-axis, opposite only the to the offset tube distance axis. Due values to the fact in thethatx the> 0tube domain weredeformation extracted and is almost the entire symmetric curves about were the mirrored y-axis, only about the theoffsety-axis distance when values needed. in the x > 0 domain were extracted and the entire curves were mirrored about the y-axis when needed. 3.1. Logistic Regression Model and Validation of Finite Element Analysis (FEA) Calculation 3.1. Logistic Regression Model and Validation of Finite Element Analysis (FEA) Calculation It is found that the outer wall profile in the necking region can be described using a logistic regressionIt model,is found asthat shown the outer in Figurewall profile5. The in redthe necking dotted lineregion in can Figure be described5a is the usi centerng a linelogistic of the neckedregression tube. The model, shapes as shown of the in tube Figure on 5. both The red sides dotted of the line center in Figure line 5a are is the symmetrical. center line of Red the necked lines shows the profiletube. The of theshape neckeds of the tube tube in on both both FEA sides and of the tensile center test line (Figure are symmetrical5a) have the. Red same lines trendshows with the the profile of the necked tube in both FEA and tensile test (Figure 5a) have the same trend with the logistic logistic regression model (Figure5b). Logistic regression was developed by statistician David Cox in regression model (Figure 5b). Logistic regression was developed by statistician David Cox in 1958 1958 [36,37]. The logistic function is shown below. [36,37]. The logistic function is shown below.
AAA1 − A2 y = 12+ A (4) yAp 22 (4) 1 + (x/x0) p 1 (xx /0 ) wherewhereA1 is A the1 is the lower lower boundary boundary of of the the curve curve andand A2 isis the the upper upper boundary boundary of the of thecurve. curve. x0 is xat0 theis at the centercenter of curvature. of curvature.p denotes p denotes the the slope slope of of thethe curve.curve.
Figure 5. Schematic diagram of logistic regression model: (a) a curve that describes the profile of outer Figure 5. Schematic diagram of logistic regression model: (a) a curve that describes the profile of outer wall; (b) mathematical model of logistic regression. wall; (b) mathematical model of logistic regression. If we select the offset distance in the y direction of outer tube wall outside the necking zone If(straight we select section the of offset the tube) distance as zero, in then the Ay2 =direction 0. The model of outer can be tube simplified wall outsideas follow. the necking zone (straight section of the tube) as zero, then A2 = 0. TheA model can be simplified as follow. y p (5) 1 (xx /0 )
where A denotes the maximum (in absolute value) offset distance of the outer wall.
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A y = p (5) 1 + (x/x0) Metals 2017, 7, 100 7 of 14 where A denotes the maximum (in absolute value) offset distance of the outer wall. Using the the l logisticogistic regression, regression, we we can can fit fit the the parameters parameters of ofA, Ax0, andx0 and p withp with test testdata. data. Table Tabless 2 and2 3and show3Metals show the 2017 the fitted, 7, fitted100 parameters parameters of of the the logistic logistic regression regression model model for for different different tub tubee sizes at7 fractureof fracture 14 moment in tension test. The The curves curves drawn using using this this model can be assumed as the actual outer wall profileprofile of Using the tubes the logistic after necking.regression, Thus, we can we fit can the parameterseasily obtain of A the , x0 andentire p with tube test surface data. Table profile profiles 2 and and the offset3 distance show the by fitted the logisticl parametersogistic regressionreg ression of the l ogistic model regression with only model a few datafordata different points.points. tube sizes at fracture moment in tension test. The curves drawn using this model can be assumed as the actual outer wall Finite element model can also predict the outer wall profile profile in the necking zone. Finite element profile of the tubes after necking. Thus, we can easily obtain the entire tube surface profile and the simulations were were conducted conducted for for the the tube tube model model OD OD0 = 6 mm= 6 with mm withdifferent different wall thickness wall thickness and WT and0 = offset distance by the logistic regression model with only0 a few data points. 1WT mm0 = with 1 mmFinite differentwith element different OD model0 values. OD can0 also values.The predict outer The thewall outer outer profiles wall wall profiles profileat fracture in at the fracture were necking extracted were zone. extractedFinite from element the from results the andresults putsimulations and together put togetherwere with conducted the with test thedata for test the and tube data logistic model and logisticfitted OD0 = curves 6 fitted mm with curvesin Figure different ins Figures 6 walland thickness7.6 and 7. and WT0 = 1 mm with different OD0 values. The outer wall profiles at fracture were extracted from the results and put together with the test data and logistic fitted curves in Figures 6 and 7.
Figure 6 6.. Outer wall profiles profiles at fracture from tensile test and simulation and the logisticlogistic regression Figure 6. Outer wall profiles at fracture from tensile test and simulation and the logistic regression fitted curves in different wall thicknesses. fittedfitted curves curves in different in different wall wall thicknesses. thicknesses.
FigureFigure 7. Inner 7. Inner wall wall profiles profiles at at fracture fracture from from tensile test test and and simulation simulation results results and and the l theogistic logistic regression fitted curves in different outer diameters. Figureregression 7. Inner fitted wallcurves profiles in different at fracture outer diameters. from tensile test and simulation results and the logistic regression fitted curves in different outer diameters. Table 2. Fitted parameters of the logistic regression model for OD0 = 6mm tubes with different WT0. The comparison shows that the outer wall profile calculated from finite element analysis (FEA) Wall Thickness (mm) Max Offset Distance of Outer Wall (mm) A (mm) x0 (mm) p correlatesTable with 2. Fitted the parameters test data and of the logistic logistic model regression very well.model The for OD maximum0 = 6mm tubes error with on Figure different6 is WT less0. than WT0 = 0.6 −0.3 −0.3 1.96 2.11 0.03 mm (when WT = 2 mm), while the maximum error on Figure7 is smaller than 0.1 mm (when Wall ThicknessWT0 = (mm) 01 Max Offset Distance−0.38 of Outer Wall (mm) −0.38A (mm) 1.93x0 (mm)2.01 p OD0 = 12WT mm).0 WT= The0.60 = 2 excellent agreement between−0.61−0.3 FEA and test data indicates−0.61−0.3 that1.95 the1.96 finite 2.24 element2.11 WT0 = 1 −0.38 −0.38 1.93 2.01 Table 3. Fitted parameters of the Logistic regression model for WT0 = 1 mm tubes with different OD0. WT0 = 2 −0.61 −0.61 1.95 2.24
Outer Diameter (mm) Max Offset Distance of Outer Wall (mm) A (mm) x0 (mm) p Table 3. Fitted parameters of the Logistic regression model for WT0 = 1 mm tubes with different OD0.
Outer Diameter (mm) Max Offset Distance of Outer Wall (mm) A (mm) x0 (mm) p
Metals 2017, 7, 100 8 of 14 model used in this study can predict the tube deformation with great accuracy, and that we can rely on the FEA results to study tube forming mechanics in detail.
Table 2. Fitted parameters of the logistic regression model for OD0 = 6mm tubes with different WT0.
Wall Thickness (mm) Max Offset Distance of Outer Wall (mm) A (mm) x0 (mm) p Metals 2017, 7, 100 8 of 14 WT0 = 0.6 −0.3 −0.3 1.96 2.11 WT0 = 1 −0.38 −0.38 1.93 2.01 OD0 = 6 −0.38 −0.38 1.93 2.01 WT0 = 2 −0.61 −0.61 1.95 2.24 OD0 = 8 −0.45 −0.45 2.22 2.02 OD0 = 10 −0.51 −0.51 3.00 2.08 Table 3. Fitted parameters of the Logistic regression model for WT0 = 1 mm tubes with different OD0. OD0 = 12 −0.57 −0.57 4.26 3.43 The comparison shows that the outer wall profile calculated from finite element analysis (FEA) Outer Diameter (mm) Max Offset Distance of Outer Wall (mm) A (mm) x0 (mm) p correlates with the test data and logistic model very well. The maximum error on Figure 6 is less than OD0 = 6 −0.38 −0.38 1.93 2.01 0.03 mm (when WT0 = 2 mm), while the maximum error on Figure 7 is smaller than 0.1 mm (when OD0 = 8 −0.45 −0.45 2.22 2.02 0 OD = 12 mm).OD0 = The 10 excellent agreement between−0.51 FEA and test data indicates−0.51 that 3.00the finite element 2.08 model usedOD in0 =this 12 study can predict the tube −deformation0.57 with great accuracy,−0.57 and 4.26 that we can 3.43 rely on the FEA results to study tube forming mechanics in detail. 3.2. Surface Profile Evolution of Inner and Outer Wall 3.2. Surface Profile Evolution of Inner and Outer Wall The effect of original tube wall thickness on surface profile at fracture moment is displayed in The effect of original tube wall thickness on surface profile at fracture moment is displayed in Figure8. Figure8a shows that when OD = 6 mm, the minimum diameter of outer wall at necking Figure 8. Figure 8a shows that when OD00 = 6 mm, the minimum diameter of outer wall at necking zonezone is inversely is inversely proportional proportional to to thethe wall thickness. thickness. In other In other word words,s, necking necking is more is prominent more prominent if the if the originaloriginal tubetube wall is thicker. The The length length of of the the necking necking zone zone increases increases very very slightly slightly when when the wall the wall becomesbecomes thicker. thicker.
Figure 8. Effect of WT0 on surface profile at fracture of (a) outer wall and (b) inner wall. Figure 8. Effect of WT0 on surface profile at fracture of (a) outer wall and (b) inner wall. The surface profile of inner wall at fracture moment in Figure 8b shows an opposite trend. The shrinkageThe surface of ID profile decreases of inneras wall wall thickness at fracture increases. moment When WT in0 Figure= 1.5 mm,8b the shows maximum an opposite offset (the trend. The shrinkageminimum ID) of IDappears decreases at x = ±0.5 as wall instead thickness of x = 0. increases. As WT0 increases When beyond WT0 = 1.5twomm, millimeters the maximum, the inner offset wall offsets negatively first and then changes direction. As a result, the final inner wall profile shows a “W” shape. When WT0 = 2.5 mm, the later expansion of ID at x = 0 is so dominant and the final IDf value here is already greater than that in the straight section of the tube (the tube outside the necking region). The expansion is about 0.05 mm.
Metals 2017, 7, 100 9 of 14
(the minimum ID) appears at x = ±0.5 instead of x = 0. As WT0 increases beyond two millimeters, the inner wall offsets negatively first and then changes direction. As a result, the final inner wall profile shows a “W” shape. When WT0 = 2.5 mm, the later expansion of ID at x = 0 is so dominant and the final ID value here is already greater than that in the straight section of the tube (the tube outside the Metalsf 2017, 7, 100 9 of 14 necking region). The expansion is about 0.05 mm. ThisThis phenomenon phenomenon can can be be explained explained byby thethe competing effects effects of ofvolume volume constancy constancy and necking. and necking. DuringDuring tension, tension, the the tube tube becomes becomes longer and and both both OD OD and and ID should ID should become become smaller smaller due to the due fact to the fact ofof volume constancy. constancy. Once Once necking necking occurs, occurs, the wall the of wall the tube of the tends tube to tendsget thinner to get rapidly. thinner Wall rapidly. Wallthinning means means OD OD becomes becomes smaller smaller while while ID becomes ID becomes larger largerand both and of both them of move them toward move the toward the “mid-layer”“mid-layer” of the wall t thickness.hickness. Both Both volume volume constancy constancy and and necking necking tend tend to make to make OD become OD become smaller. As a result, the overall shrinkage of OD is the superposition of these two effects. While for smaller. As a result, the overall shrinkage of OD is the superposition of these two effects. While for ID, the volume constancy tends to make ID become smaller. However, the necking effect will cause ID, the volume constancy tends to make ID become smaller. However, the necking effect will cause the ID become larger. In other words, the two effects are making ID moving in opposite directions. the ID become larger. In other words, the two effects are making ID moving in opposite directions. The inner wall profile (or IDf) is determined by the dominancy of these two factors. When the original The innerwall thickness wall profile is small (or ID(heref) is WT determined0 < 1.5 mm) by, the the necking dominancy effect is of not these significant. two factors. Therefore, When ID the is still original wall thicknessshrinking, isyet small the rate (here is still WT slower0 < 1.5 than mm), OD. the Once necking the original effect wall is not thickness significant. is large Therefore, enough (here ID is still shrinking,WT0 > 2 yet mm), the the rate necking is still effect slower is dominant than OD. and Once the theexpansion original rate wall is larger thickness than the is largeshrinking enough rate. (here WT0 Therefore,> 2 mm), the superposition necking effect of isthe dominant two factors and will the result expansion in the expansion rate is larger of ID. than the shrinking rate. Therefore,Figure the superposition 9 shows the effect of the of OD two0 on factors the tube’s will surface result in profile. the expansion It indicates of the ID. offset distances of both outer and inner walls reach maximum at x = 0 (both ODf and IDf reach the minimum values). Figure9 shows the effect of OD 0 on the tube’s surface profile. It indicates the offset distances of The offset distances of both outer and inner walls increase as OD0 increases. Also, the rate of increase both outer and inner walls reach maximum at x = 0 (both ODf and IDf reach the minimum values). slows down as OD0 becomes larger. When OD0 increases from 10 to 12 mm, the shrinkage of ODf is The offset distances of both outer and inner walls increase as OD0 increases. Also, the rate of increase 0.038 mm, while the shrinkage of IDf is only 0.014 mm. This means the offset distances of the inner slows down as OD becomes larger. When OD increases from 10 to 12 mm, the shrinkage of OD is wall becomes less0 affected as the tube OD0 becomes0 larger and larger. f 0.038 mm,From while results the shrinkagein Figures 8 of and ID f9,is it only can be 0.014 found mm. that This WT0 means has stronger the offset influence distances on the of offset the inner wall becomesdistance, while less affected OD0 has asa greater the tube influence OD0 becomes on the necking larger length. and larger.
Figure 9. Effect of OD0 on surface profile at fracture of (a) the outer wall and (b) the inner wall. Figure 9. Effect of OD0 on surface profile at fracture of (a) the outer wall and (b) the inner wall. 3.3. The Inner Surface Profile Change during Necking
Metals 2017, 7, 100 10 of 14
From results in Figures8 and9, it can be found that WT 0 has stronger influence on the offset distance, while OD0 has a greater influence on the necking length.
Metals3.3. The 2017 Inner, 7, 100 Surface Profile Change during Necking 10 of 14
FromFrom both experiment and and simulat simulation,ion, it it can can be be observed that that OD OD of of the the tube tube keeps decreasing decreasing asas tension continues, continues, while while the the geometrical geometrical change change of of the the inner inner tube tube wall wall is is more more complicated. complicated. PreviousPrevious discussion discussion indicates indicates that that the the offset offset of the of inner the inner tube wall tube is the wall consequence is the consequence of two competing of two compefactors:ting volume factors: constancy volume constancy and necking. and necking. A detailed A detailed finite element finite element analysis analysis of inner of wall inner profile wall profileevolution evolution during during necking necking in the tensile in the testtensile is presented test is presented next. next. FigureFigure 1010 showsshows the the finite finite element element results results of of the the profile profile of of inner inner wall wall at at five five different time time steps steps duringduring tension of a tube with OD00 = 6 6 mm, mm, WT WT00 == 2 2 mm. mm. The The time time steps steps are are selected selected at at same same intervals intervals fromfrom the beginning of neckingnecking tilltill fracture.fracture. ItIt cancan bebe seenseen thatthat atat thethe beginning beginning stage stage of of necking necking (step (step 1 1to to step step 2), 2), the the whole whole inner inner wall wall surface surface offsetsoffsets negativelynegatively (toward(toward thethe axisaxis ofof the tube). This indicates indicates thethe necking necking effect effect is is not not obvious at this time. From From step step 3, 3, the the inner inner wall close to x = 0 0 starts starts to move backwardsbackwards,, w whilehile the wall surface area around x = 1 1 continues continues to to move move negatively. negatively. Starting Starting from from this this timetime step, thethe neckingnecking effecteffect becomesbecomes dominant dominant and and the the speed speed of of wall wall thinning thinning is muchis much faster faster than than the thereduction reduction of diameter. of diameter. The The different different direction direction of motionof motion at centerat center (− 1(−1 < x<
FigureFigure 10 10.. GeometricalGeometrical change change of of inner inner tube tube wall wall during during necking necking process process..
Figure 11 shows an equivalent plastic strain contour in the quarter model of a tube with OD0 = 8 Figure 11 shows an equivalent plastic strain contour in the quarter model of a tube with OD0 = 8 mm and WT0 = 2 mm. At the beginning of necking (Status 1 and 2), the inner wall moves inwards. At mm and WT = 2 mm. At the beginning of necking (Status 1 and 2), the inner wall moves inwards. Status 1, the strain0 in the wall of the tube is still uniformly distributed in the middle section of the At Status 1, the strain in the wall of the tube is still uniformly distributed in the middle section of tube. From Status 2, higher strain starts to concentrate at a small local area, which indicates the the tube. From Status 2, higher strain starts to concentrate at a small local area, which indicates the necking effect is becoming obvious. At Status 3, the necking effect is very significant and two necking necking effect is becoming obvious. At Status 3, the necking effect is very significant and two necking bands have formed. High strain concentration is shown within the necking bands. A local area on the bands have formed. High strain concentration is shown within the necking bands. A local area on the inner wall surface where two necking bands intersect starts to move outwards. Status 3 is a special inner wall surface where two necking bands intersect starts to move outwards. Status 3 is a special moment when the offset distance of the inner wall in the middle of the necking zone is equal to that moment when the offset distance of the inner wall in the middle of the necking zone is equal to that at at the straight section of the tube. At Status 4, the ID at the same location grows larger and ultimately the straight section of the tube. At Status 4, the ID at the same location grows larger and ultimately causes fracture. causes fracture.
Metals 2017, 7, 100 11 of 14
Metals 2017, 7, 100 11 of 14
Metals 2017, 7, 100 11 of 14
Figure 11. Offset status change of the inner wall in a tube with OD0 = 8mm and WT0 = 2mm. Figure 11. Offset status change of the inner wall in a tube with OD0 = 8mm and WT0 = 2mm. 3.4. Effect of Original Diameter and Wall Thickness on the Diameter and Elongation at Fracture 3.4. Effect of Original Diameter and Wall Thickness on the Diameter and Elongation at Fracture Figure 12 shows the influence of OD0 and WT0 on the diameter of the neck at fracture, ODf. The data from FigureFEA are 11. theOffset smallest status change outer of diameter the inner wallof the in atube tube withwhen OD the0 = 8mmfracture and WTjust0 =begins. 2mm. Since it is Figure 12 shows the influence of OD0 and WT0 on the diameter of the neck at fracture, ODf. difficult to judge when the fracture will begin and stop the tension without delay during the The data from3.4. Effect FEA of Original are the Diameter smallest and outer Wall Thickness diameter on the of Diameter the tube and when Elongation the at fracture Fracture just begins. Since it experiment, ODf from tensile tests are measured after fracture. The FEA results are in good agreement is difficult to judge when the fracture will begin and stop the tension without delay during the with theFigure experimental 12 shows the data. influence Figure of 12 ODa,b0 and shows WT 0 thaton the when diameter WT0 ofremains the neck constant, at fracture, OD ODf increasesf. The experiment,linearlydata OD from withf from FEA increasing tensileare the smallestOD tests0. When are outer measured OD diameter0 remains afterof constant,the fracture.tube when ODf Thedecreasesthe fracture FEA resultslinearly just begins. areas WT inSince0 goodincreases. it is agreement difficult to judge when the fracture will begin and stop the tension without delay during the with theIn experimental Figure 12c, the tube data. diameter Figure shrinkage 12a,b shows ratio at that fracture when is defined WT0 remains as (OD0 – constant,ODf)/OD0. It OD canf increasesbe experiment, ODf from tensile tests are measured after fracture. The FEA results are in good agreement observed that when WT0 remains constant, the shrinkage ratio of outer diameter at fracture changes linearly with increasing OD0. When OD0 remains constant, ODf decreases linearly as WT0 increases. with the experimental data. Figure 12a,b shows that when WT0 remains constant, ODf increases very little and generally maintains at around 0.31. And in Figure 12d, when OD0− remains constant, In Figure 12linearlyc, the with tube increasing diameter OD shrinkage0. When OD0 ratioremains at constant, fracture OD isfdefined decreases aslinearly (OD 0as WTOD0 increases.f)/OD0 . It can be the shrinkage ratio increases linearly with WT0. This is why ODf is linearly proportional to OD0 and observed thatIn Figure when 12c, WT the 0tuberemains diameter constant, shrinkage the ratio shrinkage at fracture is ratio defined of outeras (OD diameter0 – ODf)/OD at0. It fracture can be changes is inversely linearly proportional to WT0. observed that when WT0 remains constant, the shrinkage ratio of outer diameter at fracture changes very little and generally maintains at around 0.31. And in Figure 12d, when OD0 remains constant, the very little and generally maintains at around 0.31. And in Figure 12d, when OD0 remains constant, shrinkage ratio increases linearly with WT0. This is why ODf is linearly proportional to OD0 and is the shrinkage ratio increases linearly with WT0. This is why ODf is linearly proportional to OD0 and inversely linearlyis inversely proportional linearly proportional to WT to0. WT0.
Figure 12. Effect of original diameter and wall thickness on tube diameter at fracture: (a) Effect of original diameter on fracture diameter; (b) Effect of original wall thickness on fracture diameter; (c) Effect of original diameter on diameter shrinkage; (d) Effect of original wall thickness on diameter shrinkage. Metals 2017, 7, 100 12 of 14
Figure 12. Effect of original diameter and wall thickness on tube diameter at fracture: (a) Effect of Metals 2017original, 7, 100 diameter on fracture diameter; (b) Effect of original wall thickness on fracture diameter; (c) 12 of 14 Effect of original diameter on diameter shrinkage; (d) Effect of original wall thickness on diameter shrinkage. In the same way as above, the elongation at fracture of the gauge section calculated from experimentIn the and same finite way element as abov simulation,e, the elongation respectively, at fracture are plotted of the gauge in the section Figure calculated13. It reveals from the effectexperiment of OD0 andand WTfinite0 onelement tube elongationsimulation, atrespectively, fracture. The are data plotted from in tensilethe Figure tests 13. are It always reveals athe little biteffect larger of than OD0 thoseand WT from0 on FEA;tube elongation one of the at reasons fracture. should The data come from from tensile the tests measure are al moment.ways a little We bit get thelarger gauge than elongation those from at FEA fracture; one onceof the the reasons fracture should begins come in from FEA the but measure after fracture moment. in We tensile get the tests. Thegauge average elongation error is at about fracture five percent,once the butfracture the curves begins have in FEA the but same after trend. fracture In Figure in tensile 13a, astests. OD The0 gets 0 larger,average tube error elongation is about increases five percent slightly., but Inthe Figure curves 13 hab,ve as the WT same0 getting trend. larger, In Figure tube 13 elongationa, as OD almostgets larger, tube elongation increases slightly. In Figure 13b, as WT0 getting larger, tube elongation almost keeps the same value. The elongation becomes more affected by OD0 and less affected by changes in keeps the same value. The elongation becomes more affected by OD0 and less affected by changes in WT0. It can be inferred that the wall thickness has almost no impact on the elongation at fracture in a small-diameterWT0. It can bethin-walled inferred that 1Cr18Ni9Ti the wall thickness tube. has almost no impact on the elongation at fracture in a small-diameter thin-walled 1Cr18Ni9Ti tube.
FigureFigure 13. 13Effect. Effect of of (a ()a original) original diameter diameter andand ((bb)) originaloriginal wall wall thicknes thicknesss on on the the elongation elongation at atfracture fracture..
4. Conclusions 4. Conclusions A large number of tensile tests of 1Cr18Ni9Ti tubes with different sizes were conducted. Finite A large number of tensile tests of 1Cr18Ni9Ti tubes with different sizes were conducted. element method demonstrated its advantages in studying the tube forming mechanics. Surface Finite element method demonstrated its advantages in studying the tube forming mechanics. profiles of the tube walls from finite element simulation can match the test data with high accuracy. Surface profiles of the tube walls from finite element simulation can match the test data with high Combined with a large number of experiments and FEA results, the following conclusions can be accuracy.made. Combined with a large number of experiments and FEA results, the following conclusions can be made. (1) The geometry of outer tube wall in the necking region can be described using a logistic (1) Theregression geometry curve. of outer Thus tube, the wall mathematical in the necking formula region of necked can be describedtube surface using profile a logistic and the regression offset curve.distance Thus, can the be mathematicalobtained by the formula calculated of neckedlogistic tuberegression surface model profile with and only the a offsetfew data distance points. can beThe obtained formula by could the calculated be used in logistic combination regression with modelBridgman’s with equation only a few in datathe future points. to Thederive formula the couldpost- benecking used instress combination–strain relationships with Bridgman’s of tubes. equation in the future to derive the post-necking (2) stress–strainDuring the relationships uniaxial tension of tubes. of tube, both OD and ID have a shrinking trend, though the shrinking rates are different. The offset distance of outer tube wall increases as the original wall (2) During the uniaxial tension of tube, both OD and ID have a shrinking trend, though the shrinking thickness or outer tube diameter increases. The offset distance of inner tube wall decreases as rates are different. The offset distance of outer tube wall increases as the original wall thickness the original wall thickness increases. If the wall of the tube is thick enough, the final inner wall or outer tube diameter increases. The offset distance of inner tube wall decreases as the original will expand instead of shrinking. wall thickness increases. If the wall of the tube is thick enough, the final inner wall will expand (3) The offset (shrinking or expansion) of OD and ID are affected by two competing factors: volume insteadconstancy of shrinking. and necking. For the case of OD, both factors will cause the OD shrink. So OD shrinks (3) Thefaster offset when (shrinking necking orbecomes expansion) dominant. of OD However, and ID are for affected the case by of twoID, the competing two factors factors: will cause volume constancythe ID to move and necking. in opposite For directions. the case The of OD,final bothmotion factors of the willID is causedetermined the OD by shrink.the dominant So OD shrinksfactor. faster when necking becomes dominant. However, for the case of ID, the two factors will cause the ID to move in opposite directions. The final motion of the ID is determined by the dominant factor. Metals 2017, 7, 100 13 of 14
(4) The final geometry of the tube is determined by the original outer tube diameter and wall thickness. The offset distances of outer and inner walls are mainly affected by the original wall thickness. The length of the necking zone is more influenced by the original outer tube diameter. (5) Tube outer diameter at fracture moment is linearly proportional to original diameter and is inversely linearly proportional to original wall thickness. Because when original wall thickness remains constant, the outer diameter shrinkage ratio at fracture remains almost unchanged and when original diameter remains constant, the shrinkage ratio increases linearly with original wall thickness. As tube diameter getting larger, the tube elongation increases slightly. The wall thickness has almost no impact on the elongation at fracture in small-diameter thin-walled 1Cr18Ni9Ti tube.
Acknowledgments: The work described in the paper was part of research within the program Grant No. 51175044 from National Natural Science Foundation of China under. The authors are very grateful for the financial support. Author Contributions: Chong Li and Jingwen Zhang conceived and designed the experiments; Chong Li, Ning Yi and Jingwen Zhang performed the experiments. Chong Li performed the finite element simulations, analyzed the date and wrote the manuscript. Daxin E supervised the whole study. All authors reviewed the final paper. Conflicts of Interest: The authors declare no conflict of interest.
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