RESEARCH PAPER

Inferring Post-Necking Strain Hardening Behavior of Sheetsby a Combination of Continuous Bending Under Tension Testingand Finite Element Modeling

C.M. Poulin1& T.J. Barrett1 & M. Knezevic1

Received: 26 May 2019 /Accepted: 30 December 2019 /Published online: 9 January 2020# Society for Experimental Mechanics 2020

AbstractThis paper presents a combined experimental and simulation approach to identify post-necking hardening behavior of ductilesheet metal. The method is based on matching a measured load-displacement curve from a continuous bending under tension(CBT) test with the curve simulated using the finite element method (FEM), while adjusting an input flow stress curve into theFEM. The CBT test depletes ductility uniformly throughout the gauge section of a tested sheet, and thus, stretches the sheet farbeyond the point of maximum uniform strain in a simple tension (ST) test. Having the extended load-displacement curve, thecalibrated flow stress curve is extrapolated beyond the point of necking. The method is used to identify the post-neckinghardening behavior of an aluminum alloy, AA6022-T4, and two dual-phase (DP) steels, DP 980 and DP 1180. One measuredload-displacement curve is used for the identification of a flow stress curve per material, and then the flow stress curve is used tosimulate two additional measured load-displacement curves per material for verification. The predictions demonstrate the utilityof the developed CBT-FEM methodology for inferring the post-necking strain hardening behavior of sheets. Furthermore, theresults for AA6022-T4 are compared with the hydraulic bulge test data. Unlike the hydraulic bulge test, the CBT-FEM methodcan predict increasing as well as decreasing anisotropic hardening rate in the post-necking regime. The latter is associated withprobing stage IV hardening. The proposed methodology along with the results and these key advantages are presented anddiscussed in this paper.

Keywords Plasticdeformation .Post-neckingstrainhardening .Continuousbendingunder tension .Finiteelementmethod .Sheetmetal

Introduction

Accurate characterization of a material stress-strain responseis crucial to accurately simulate forming processes using thefinite element method (FEM). During forming operationsstrain levels greater than those corresponding to a maximumuniform elongation are often generated [1–3]. The simple ten-sion (ST) test is a ubiquitous experiment used to characterizethe stress-strain response of a material due to readily availabletesting equipment and simplicity. The test allows for directmeasurement of the hardening curve up to the point of diffuse

necking. Post-necking behavior is often estimated using phe-nomenological hardening laws producing different results de-pending on the hardening law selected to fit the availableexperimental pre-necking data and then to approximate thepost-necking region.

As opposed to fitting phenomenological approxima-tions, alternative methodologies have been developed toidentify post-necking hardening behavior of materials.The early work in this field has been described in [4].The idea was to analytically correct and compensate forstress-strain fields based on the measurement of evolvinggeometrical features of the necking profile, which re-quired a significant experimental effort. More recently,digital image correlation (DIC) technique for measure-ment of strain fields has made the experimental effortsto measure the strain fields in the diffuse neck much moreconvenient. DIC measurements have been extensivelyused to study the hardening behavior in the post-necking

* M. Knezevicmarko.knezevic@unh.edu

1 Department of Mechanical Engineering, University of NewHampshire, 33 Academic Way, Kingsbury Hall, W119,Durham, NH 03824, USA

Experimental Mechanics (2020) 60:459–473https://doi.org/10.1007/s11340-019-00577-1

regime in combination with FEM simulations [5–8]. Tothis end, the FEM was used to iteratively model tests,while the material parameters are calibrated. The calibra-tion involves a minimization of a selected objective func-tion, which is usually set as the difference between themeasured and calculated displacement or strain fields. Themethodology is referred to as the inverse FEM. In sum-mary, with the aid of DIC to make local measurements ofthe displacement field and optimization tools, the materialmodel calibration into the post-necking range can be per-formed. Nevertheless, there are several issues in applyingit to the problem of diffuse necking. In particular, a loca-tion of the diffuse neck in a tensile sample is challengingto predict by standard FEM. In order to control the local-ization location in an FEM simulation, an artificial notchhas to be introduced. Introducing a notch often createsnumerical difficulties. To relax the issue of controllingthe location of the diffuse neck, the use of tapered sam-ples has been proposed [9]. Additionally, due to typicallyvery large plastic strains developing at the localization, aproper mesh design is very importance to simulate thenecking profile and underlying mechanical fields. Thusbuilding a reliable FEM model capable of dealing withthe plastic instability is challenging.

Another method based on the same kind of experi-ments but different calculations to identify the post-necking hardening behavior of sheet metals is based onthe minimization of the discrepancy between the internal(calculated) and external (measured) work in the neckingzone during a ST test. The method is known as the virtualfields method (VFM) [10–14]. The method is restricted tothin sheets but circumvents the deficiencies of the FEMinverse method. Conceiving independent tests capable ofprobing post-necking material behavior under ST is chal-lenging due to plastic instabilities. The VFM method hasbeen experimentally verified in the post-necking regimeusing an independent tube expansion specialized testingsetup demonstrating good agreement [15]. The tube ex-pansion test expands a tubular specimen under ST.Combining an axial load and internal pressure deformssamples in ST to higher strains than without the pressure.Another test to measure the stress strain response athigher strains is the bulge test [16, 17]. In this test, hy-draulic pressure is applied to a circular blank of materialsubjecting it to an approximately equibiaxial stress state.Based on the pressure applied, curvature of the bulgedspecimen, and thickness at the pole, the biaxial stress-strain response can be measured. The test provides astrain hardening curve which is similar to a ST curvebut is ex t rapola ted to la rge s t ra in levels [10] .Comparison between the hydraulic bulge test and the tubeexpansion test revealed some discrepancies in the post-

necking regime [15]. The tube expansion test was foundto provide a faster decreasing hardening rate in the post-necking regime, which is more likely material behaviorfor the stage IV hardening [18–20].

Continuous bending under tension (CBT) has beenestablished as an experiment capable of significantly in-creasing elongation to fracture (ETF) in sheet metals [21].The test resembles closely a classical ST test while sup-pressing the instability that limits ST. During CBT testing,a sheet/strip is pulled in tension and continuously bent/unbent by rollers, which incrementally deform the mate-rial. Significantly, the CBT test was found capable ofachieving over five times improved ETF over ST fordual-phase (DP) steels [22]. It has been shown that thestrain path during CBT is very close to ST [23], making ita suitable experiment to extract hardening behavior pastthe limit of ST. For this purpose, an interrupted CBTtesting method has been developed [24]. In this method,a set of sheets/strips undergoes CBT to variable numberof cycles, and then a sub-size tensile specimen is ma-chined out of each pre-deformed sheet/strip. These speci-mens are then tested in ST and their yield strength spreadout based on the levels of accumulated effective strainachieved during CBT. Since the achieved effective strainlevels in CBT are much higher than those at the point ofmaximum uniform strain in ST, the underlying flow curveis extrapolated to high strain levels. While successful forsome materials, the approach is not general enough be-cause it breaks down for materials hardening differentlyunder CBT than ST. Enhanced strengthening of alloyAA6022-T4 under CBT relative to ST has been observedand rationalized based on microstructural characterization[25, 26]. If the strength is driven by achieved heteroge-neous microstructure meaning that the achieved effectivestrain level is insufficient to characterize the state of thematerial then the strength measured by CBT may not be areliable way to extrapolate stress-strain curves.

With the above background, this paper presents a newmethod to infer post-necking material behavior based onCBT testing. Considering that CBT testing can be suc-cessfully simulated using FEM [22, 27], the proposedmethod is a synergy between the CBT experiment andFEM simulations of the same test. The combined CBT-FEM method first matches the CBT measured and FEMsimulated force-displacement curves up to the maximumequivalent plastic strain seen in ST. From there, the post-necking hardening curve is adjusted until the entire CBTmeasured force-displacement curve is matched with theFEM simulated curve. Although the proposed method in-volves iterative FEM simulations, it does not require con-trolling the location of the diffuse neck by introducing anotch or a specialized mesh design. Thus, the iterative

460 Exp Mech (2020) 60:459–473

methodology for inferring a flow curve using the CBT-FEM method is relatively fast. The materials studied hereare an aluminum alloy, AA6022-T4 [28, 29], and two DPsteels, DP 980, and DP 1180. The imposed structural ef-fects of the CBT test facilitated stretching of these mate-rials to large strains in order to study their post-neckingbehavior. The comparisons between measured and simu-lated results show that the developed CBT-FEM methodcan determine the post-necking strain hardening behaviorof these materials. The results for AA6022-T4 are furthercompared with those of a hydraulic bulge test. The factthat that each of these materials exhibit unique post neck-ing hardening behavior, AA6022-T4 exhibits Voce-likebehavior [16, 30], while DP steels resemble Swift-likebehavior [12, 31], demonstrates generality of the CBT-FEM method. It is also shown that the method can predictincreasing as well as decreasing anisotropic hardeningrate in the post-necking regime. In closing, we discussthe origin of the load magnitude in CBT testing underdifferent structural conditions defined by crosshead veloc-ity and bending depth. These insights were facilitated bythe comprehensive set of simulations and experiments car-ried out in the present work.

Materials

The materials studied in this work are 1 mm thick sheets ofAA6022-T4 [29, 32], DP 980, and DP 1180 [33]. Tables 1 and2 show their chemical composition. DP 980 is fromUS Steels’hot dip (HD) processing line, while DP1180 is from theircontinuous annealing line (CAL). Additionally, DP 980 wasgalvanized (HDGI) coated, while and DP 1180 was bare.Figure 1 shows the behavior of the materials in ST. The testingprocedures are detailed in [34–38]. Specimens of DP 980sheet are tested here along the rolling direction (RD) andtransverse direction (TD) showing a small plastic anisotropy,while the specimens of AA6022-T4 and DP 1180 are testedonly along RD and TD, respectively. The data for AA6022-T4and DP 1180 is taken from our earlier works [22, 39]. Forsteels, the curves terminate at strains of a few percent due tonecking. The steels exhibit high strength and high stiffness butsmall ductility, while the alloy AA6022-T4 is moderately duc-tile but much softer. The plasticity in ST of the studied spec-imens is limited by necking, followed by shear localization,and finally fracture, while the remainder of specimens’ gauge

section away from the localized zone is deformed uniformlywith a substantial amount of remaining ductility [3]. The CBTtest exploits ductility of the entire gauge section. Unlike amajority of the available methods for extracting material be-havior at large plastic strains, the proposed method based onCBT does not focus on the necking region but considers theentire gauge section.

CBT Testing

Universal tensile testing machines furnished with a set ofmoving rollers have been typically used for CBT testing [40,41]. A specialized CBT machine has been recently presentedin [25, 42]. This machine is used in the present work (Fig. 2).The machine features a moving carriage holding a specimenand the axial loading system, a stationary roller assemblymounted on the machine base, and a data acquisition andcontrol hardware/software. The carriage reciprocates duringthe CBT test with a velocity of 66 mm/s. The rollers are ofD = 25.4 mm diameter providing a bending strain of 0.038 fora sheet of 1 mm if wrapping around is ensured. The bottomrollers are L = 54 mm apart, while the top roller is verticallyadjustable providing a possibility to set a bending depth, δ.The machine has two donut-style Futek load cells, LCF 450(capacity of 22.24 kN) attached to the carriage and LTH 500(capacity of 22.24 kN) attached to the actuator. The hydraulicactuator supplying crosshead (pulling) velocity has a BalluffMicropulse BTL7-A501-M0305-Z-S32 position sensor with aresolution of 5 μm. The machine is designed for the accuracyin terms of force and displacement measurements of 10 N and5 μm, respectively. The carriage motion is determined by apair of limit switches, one positioned on the grip attached tothe hydraulic cylinder and another positioned on the base.Limit switches signal the end of the stroke and reversion forthe moving carriage. More details in terms of the machinedesign can be found in [39, 42].

Figure 3(a) shows a drawing of a CBT sample used in thepresent work for all three materials. The deformation regionsdenote the number of bendings for the sheet during each CBTpass. The sheet only undergoes one bending/unbending (1x)per pass near the grips. The sheet is bent/unbent three times(3x) in the center of the sample. The gauge length of 200 mmis chosen such that the rollers never exit the gauge section of astandard ASTM E8 tensile specimen while producing the 1x,2x, and 3x deformation regions. The setup with limit switchesfacilitates every pass/stroke to be longer increasing the 3xdeformation region, while keeping the 1x and 2x regions con-stant.We will refer to rollers traversing the entire gauge lengthin one direction to the other side as a CBT pass, while rollerstraversing the gauge length to the other side and then backover the gauge section to the original side as a CBT cycle. Inthis work, the testing of DP 980 has been carried out, while the

Table 1 Chemical composition of alloy AA6022-T4 (wt%)

Si Fe Cu Mn Mg Cr Zn Ti Tin Al

0.90 0.10 0.045 0.053 0.57 0.027 0.016 0.025 <0.02 balance

Exp Mech (2020) 60:459–473 461

data for AA6022-T4 and DP 1180 is taken from our earlierworks [22, 39]. These testing results will be presented alongwith the simulation results.

CBT-FEM Method

The CBT testing described in section 3 is simulated numer-ically using the nonlinear FEM solver in Abaqus.Figure 3(b) shows the finite element model and mesh. The

boundary conditions are imposed to be consistent with ex-periments. Vertical motion of the top roller determinesbending depth, while crosshead velocity is directly appliedto the sample. The gripped regions are modeled as analyti-cally rigid, with one end held stationary and the otherdisplaced at a rate in accordance with the experimentallyrecorded data. Contact between the rollers and the specimenis enforced using the default state-based contact formulationto ensure that nodes do not penetrate into the rollers. Afriction coefficient of 0.17, which was measured by CBTin [42], is used for all simulations. Bending is initiated bydisplacing the middle roller into the specimen by apredetermined amount measured carefully using gaugeblocks in the experimental configuration. Additionally, theroller velocity profile is taken directly from the experimentsand imported into the simulations making the experimentsand simulations one-to-one. Because of the symmetry, theFE model is reduced in half and consists of 9828 C3D8R(continuum 3D eight node reduced integration elements).The number of elements across the thickness is 4, with thein-plane element size of 1 mm along the length by 0.65 mmin the width. A mesh-sensitivity study was performed. Anyfurther refinement of the mesh by adding more elements leftthe results unchanged and only increased the computationaltime involved. The constitutive law in the simulations is therate-independent J2 plasticity theory of von Mises. TheMises yield surface expands isotropically with a truestress-true strain curve, which is provided as an input intothe simulations [43, 44]. This input true stress-true straincurve is iteratively calibrated to large plastic strains sincethe strain levels in CBT go beyond those in ST. As a result,the curve is extrapolated, while matching the CBT load-displacement data. Details pertaining to the extrapolationprocedure are given in Appendix.

Results

Figure 4 shows calculated axial strain (LE11) distributionin a specimen of AA6022-T4 during CBT under a bend-ing depth of 2.5 mm and a crosshead velocity of 1.2 mm/s. As can be seen, a small amount of strain is introduced

Fig. 1 Engineering (dashed lines) and true (solid lines) stress-straincurves for (a) AA6022-T4 along the rolling direction (RD) and (b) twosteels along the transverse direction (TD) for DP 1180 and along both RDand TD for DP 980 measured in simple tension (ST) under a strain rate of0.001 s−1

Table 2 Chemical compositionof DP steels (wt%) C Mn P S Si Cu Ni Cr Mo

DP 980 0.11 2.411 0.013 0.005 0.013 0.027 0.009 0.255 0.385

DP 1180 0.168 2.222 0.015 0.0053 1.421 0.021 0.007 0.036 0.013

Sn Al Zr V Cb Ti B N2

DP980 0.006 0.049 0.005 0.011 0.004 0.002 0.0001 0.0033

DP1180 0.007 0.051 0.005 0.012 0.007 0.039 0.0004 0.0086

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Fig. 2 (a) A photograph showingthe main components of the CBTtesting machine. (b) Schematic ofthe CBT process and definition ofkey components: bending depth,δ, achieved by adjusting the toproller vertically, sheet thickness, t,roller diameters, D = 25.4 mm,and distance between the bottomrollers, L = 54 mm

Fig. 3 (a) Drawing of a CBTsamplewith dimensions in [mm]. The same sample dimensions are used for samples of AA6022-T4, DP 1180, and DP 980.Also, the initial regions, 1×, 2×, and 3×, are indicated. (b) ½ FEmodel consisting of 9828 C3D8R elements used for the CBTsimulations in views tiltedat 45°

Exp Mech (2020) 60:459–473 463

upon lowering of the top roller (Fig. 4(b)). After the si-multaneous application of pulling, the strain develops rel-atively uniformly throughout the 3x deformation regionand accumulates with CBT cycling. Figure 4 shows thestrain fields from the upper surface. The highest level ofstrain (the hot spot) is between 2x and 3x regions, whichhas been attributed to the geometrical discontinuity be-tween the two regions [22]. The strain in the 2x and 1xregions is much smaller. As is evident from Fig. 4(g), thegauge section exhibits relatively uniform strain across the

thickness. The strain is not uniform underneath the rollers,as will be shown later in Fig. 8. The contour maps like inFig. 4 can be plotted for every simulation case. Due totheir similarity, the maps are shown only for the firstsimulation case (AA6022-T4 under 2.5 normalized benddepth and 1.2 mm/s crosshead speed).

Table 3 presents the axial strain levels in 3x predicted withthe CBT cycles per material. As these values are taken fromthe center of the samples (i.e. the middle of 3x deformationregion) where the strains are uniform, the hot spot strain levels

Fig. 4 LE11 contour mapsdeveloped during the CBT testsimulated in Abaqus: (a) beforeroller engagement, (b) after rollerengagement, before tension, (c)after 2 CBT cycles, (d) after 4CBT cycles, (e) after 6 CBTcycles, and (f) at fracture. (g)Strain distribution across the sheetthickness at the end of each cycle

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are slightly higher. As is evident, these strain levels in CBTaremuch higher than those achieved in ST. The flow stress curveper material must be provided to cover the correspondingstrain levels. Provided the flow stress curves, the load-displacement curves during CBT can be calculated. Figure 5shows comparisons of simulated and measured load-displacement curves during CBT for the three studied mate-rials after their adjustment. Load-displacement curves for STare also shown to recognize the amount of enhanced elonga-tion in CBT.

The structural effects of the CBT testing result with theload-displacement curves exhibiting spikes. Each stroke be-gins with the carriage accelerating to a constant velocity. Theconstant velocity (i.e. the steady state plateau) lasts for themajority of the stroke and then the carriage decelerates andstops. The same repeats in every stroke/pass. The spikes in theforce profile form during the acceleration and deceleration. Atthe stop, there is no bending. As a result, the sample reachesthe ST condition raising the force in order to facilitate anyplastic flow in ST because the pulling does not stop. As thecarriage starts moving again, now in the opposite direction,

the CBT continues dropping the force. The height of thespikes is a function of the bending amount, as will be evidentfrom more tests and simulations presented in the next section.The slight difference between the force values for every otherplateau is a consequence of material either moving towards(lower plateau force) or away from (higher plateau force) thegiven load cell recording the data. The load levels aregoverned by the material hardening and changes in thecross-sectional area (i.e. width and thickness) of the elongat-ing specimen. Table 4 presents comparisons between mea-sured and predicted sheet width and thickness in the 3x defor-mation region before testing and after the testing for the testspresented in Fig. 5. The measurements were performed usinga micrometer at several locations. The variation between themeasuring points was small, within 3%. Average values arecalculated and presented in the table. The table shows percent(%) differences in the cross-sectional area between unde-formed CBT samples and the FE model. The initial FE modelsetup is the same for all materials and tests. The difference issmall. Furthermore, the table shows the same quantities afterthe experimental testing and simulations to the same deforma-tion condition per sample. As is evident, the difference persample has only slightly evolved meaning that the simulatedand measured cross-sectional areas are in excellent agreement.Accordingly, the extracted hardening response for the samplescan be regarded as accurate.

In order for the measured and simulated load-displacementto match, the input flow stress curve in the FEM simulationmust be appropriately adjusted. The adjustment of the flowstress curve is in the core of the combined CBT-FEM methodfor inferring the hardening behavior of sheets at high strainlevels. First, the CBT measured and FEM simulated force-displacement curves is matched up to the maximum equiva-lent plastic strain using the measured ST curve per material.From there, the post-necking hardening curve is adjusted untilthe entire CBTmeasured force-displacement curve is matchedwith the FEM simulated curve. The output curves for thestudied materials are shown in Fig. 6. The flow curve forAA6022-T4 is also compared with the curve measured usingthe hydraulic bulge test. The bulge test curve is taken from[16]. The alloy tested in [16] is slightly softer so a shifted

Fig. 8 Segment of axial strain (LE11) evolution at the top, middle, andbottom locations at the center of the 3x deformation region along thelength and mid-width of the CBT specimens during the three CBTprocess simulations of steel DP 980. The symbols denote the beginningand the end of the bending operations

Table 3 LE11 accumulated with the CBT cycles from the simulations in Fig. 5

# of CBT cycles 1 2 3 4 5 6 7 8 9 10 11 12

AA6022-T4 0.043 0.082 0.119 0.157 0.195 0.234 0.272 0.311 0.32

DP 980 (RD) 0.035 0.084 0.128 0.175 0.228 0.27 0.315 0.37 0.407 0.454 0.498 0.541

DP 980 (TD) 0.034 0.084 0.139 0.181 0.229 0.275 0.32 0.369 0.419

DP 1180 0.043 0.089 0.133 0.179 0.224 0.268 0.313 0.357 0.382

These values are taken from the central region of specimens i.e. the middle of the 3x region, where the strains are approximately uniform. The last (italic)value per material is a strain at fracture, which occurred before completing the given cycle

Exp Mech (2020) 60:459–473 465

curve for 22 MPa is also plotted. It can be seen that, like thetube expansion test, the CBT-FEM method also provides afaster decreasing hardening rate in the post-necking regimethan the bulge test. Importantly, the CBT-FEM method

facilitates flow stress extrapolation along any tested sampledirection, which means that it can characterize anisotropy athigh strain levels. The results for DP 980 are presented for RDand TD sample directions. In a reasonable agreement with our

Fig. 5 Comparison of simulatedand measured force-displacementcurves during CBTwithparameters indicated in thelegends for (a) AA6022-T4 alongRD, (b) DP 980 along RD, (c) DP980 along TD, and (d) DP 1180along TD. Force-displacementcurves for simple tension areprovided as references. The insertin (a) schematically shows thecorrelation between the carriagevelocity (green) and recordedforce (blue) profiles for one passduring CBT

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extrapolation for DP steels, approximately linear hardeningwith large strains has been observed for several DP steels in[45].

Discussion

This work presents a systematic methodology based on a com-bination of experiments and FEM analysis to extract the

material response at large plastic strains. Such methodologyis needed because many sheet metal forming operationsachieve high levels of plastic strain. To simulate those opera-tions, the material behavior at the required high strain levelsmust be known. The proposed methodology is referred to asthe CBT-FEM method and involves simulating a CBT testusing the FEM, while the material response is iteratively ad-justed until the calculated and measured force-displacementmatch. The extrapolated true stress-true strain curves are ob-tained for three materials. The curve for AA6022-T4 is com-pared with the bulge test data. Furthermore, the obtainedcurves are used to simulate additional CBT tests for verifica-tion. These verification tests involved a different combinationof bend depth and pulling speed, which influence the forcemagnitudes. The predictions are shown in Fig. 7. With lower-ing the bend depth, the mechanical fields become closer to ST.Reducing the speed, lowers the force, which will be rational-ized shortly. Nevertheless, the simulated load-displacementcurves are in excellent agreement with measured curves evenunder these modified CBTstructural conditions. These predic-tions imply that the CBT-FEM method can be used to extractpost-necking material behavior. Significantly, the CBT-FEMmethod can facilitate determining the evolution of anisotropyto large plastic strains because it can be applied in any sampledirection. Table 5 shows a comparison between measured andsimulated cross-sectional geometry for the cases presented inFig. 7.

Although the proposed method involves iterative FEMsimulations, it does not require controlling the location ofthe diffuse neck by introducing a notch or a specialized meshdesign. Thus, the iterative methodology for inferring a flowcurve using the CBT-FEM method is relatively easy and fast.While there is room to further improve the optimization pro-cedure, the flow curves inferred in this paper using the proce-dure presented in Appendix took about 10 days of calculations

Table 4 Initial and final (after CBT) cross-sectional geometry for samples presented in Fig. 5 to facilitate the comparison between measurements andsimulations

Specimen Initial (before testing) Final (after CBT testing to fracture)

Experiment % differencein area

Experiment Simulation % differencein area

Width[mm]

Thickness[mm]

Area[mm2]

Width[mm]

Thickness[mm]

Area[mm2]

Width[mm]

Thickness[mm]

Area[mm2]

AA6022-T4 2.5,1.2 mm/s

11.76 1.030 12.113 0.029 10.89 0.851 9.078 10.34 0.878 9.266 0.020

DP 980 (RD) 3.5,1.35 mm/s

11.74 1.010 11.857 0.010 9.95 0.740 7.363 9.36 0.770 7.207 0.021

DP 980 (TD) 3.5,1.35 mm/s

11.72 1.010 11.837 0.007 10.38 0.787 8.173 9.87 0.823 8.123 0.006

DP 1180 3.5,1.35 mm/s

11.77 1.048 12.334 0.047 10.34 0.860 8.892 10.09 0.840 8.477 0.047

The initial width and thickness of the strip used in all simulations was 11.75 mm and 1 mm, respectively. ‘% difference in area’ is the % differencebetween measured and simulated areas

Fig. 6 True stress-strain curves extrapolated from those presented inFig. 1 to facilitate simulating the load-displacement curves under CBTin Fig. 4. Bulge test experimental data is taken from [16]

Exp Mech (2020) 60:459–473 467

on a computer workstation with the following characteristics:Intel® Xeon® Gold 6154 CPU @3.00 GHz using 32 CPUs.The accuracy of the method can potentially be improved byinvolving advanced yield functions [46–49] or crystal plastic-ity constitutive descriptions [50–53] to more accurately simu-late the test. Simultaneously, the CBT-FEM method can facil-itate calibrating the evolution of anisotropy to large plasticstrains for advanced plasticity models and crystal plasticity

frameworks. Kinematic and isotropic hardening materialmodels are giving similar simulation results for the CBT sim-ulations [27].

In the remainder of the discussion section, we describe theorigin of the load magnitude variations observed as a functionof the CBT structural conditions. The simulations performedunder three different CBTconditions allow us to discuss theseeffects on load levels. The combination of a bending stresswith a tensile force in a CBT setup leads to a non-symmetricstress profile of tension/compression, with the neutral axiscyclically shifting towards the surface of the roller over whichthe sheet is passing. The addition of bending tension to thetensile load means that a lower force is required to reach theplastic flow stress of the material on the outward edge of thestrip (relative to a roller). Hence the load/extension curve issignificantly lower than a simple tensile test. Indeed, CBTprocessing is conducted at a uniaxial stress level below theyield stress of the material. In addition to the influence ofbending, the Bauschinger effect is present due to the reverseloading as a point of the strip moves from bending in tension(with, e.g., a roller above it) to bending in compression (with,e.g., a roller below) in a cyclic fashion. This also serves toreduce the required load for plastic deformation.

The load-displacement curves presented in this work havebeen obtained under three different combinations of the CBTparameters, the crosshead pulling velocity and bending depth.As already explained, the material bent more with a higherbending depth requires a lower force for the elongation. Atlower crosshead velocity the number of bending/unbendingoperations to obtain a certain elongation is higher, while theobserved force is lower. Thus, the deforming cross-sectionalstress profile of tension/compression spanning over the gaugesection of the specimen is more frequent. The lower force is adirect consequence of the lower strain increment, which isdefined using Δε ¼ vCH

vR, as will be explained shortly.

Certainly, the lower crosshead velocity produces a lower strainincrement per pass/stroke at the given bending depth resultingin a lower pulling force. In contrast, at very high crossheadvelocities, the number of bending/unbending operations is justa few. At these velocities the test is approaching ST, while therecorded force is only slightly lower than when tested withoutrolls. At very high pulling velocities, the test is no longer aCBT test because the elongation might not be happening onlywithin the zone of bending.

As the portion of the strip/sheet being bent only deformsimplies that only the material visited by the rolls elongatesdefining the gauge length of the CBT specimen. As a result,the average strain increment per stroke/pass, Δε, is the ratiobetween the elongation of the specimen as a consequence ofthe hydraulic cylinder (or crosshead) tension and the gaugelength that carriage (or roller) delineates. Thus, the strain in-crement can be estimated based on the hydraulic cylinder

Fig. 7 Comparison of simulated andmeasured force-displacement curvesduring CBTwith parameters indicated in the legends used for verificationof the extrapolated true stress-true strain curves for (a) AA6022-T4 alongRD, (b) DP 980 along TD, and (c) DP 1180 along TD

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tension (or crosshead) velocity, vCH, and the constant carriage(or roller) velocity of 66 mm/s, vR as Δε ¼ vCH

vR. Since the

carriage accelerates and decelerates during a portion of thestroke, the estimate is regarded as the lower bound.Furthermore, since there are three rollers and six bending/unbending operations, the strain increment in each bendingoperation is approximately Δε

6 .When a strip is pulled over a radius under high tension it

is safe to assume that the bending radius of the strip is moreor less equal to the die radius. However, in the CBT testthat is not the case. The pulling force can be quite low,meaning that it is not safe to assume that the strip is pulledtight against the rolls. The numerical modeling carried outin the present work further reveals the evolution of thestrain distributions through the thickness of the specimen.

In particular, it can evaluate the accuracy of the strain in-crement estimated using Δε ¼ vCH

vR. Figure 8 presents a seg-

ment of axial strain (LE11) evolution at three interrogationpoints located at the top, middle, and bottom surfaces at thecenter and mid-width of the CBT specimens during thethree CBT process simulations of steel DP 980. The strainaccumulates in a staircase fashion, with a pattern of threespikes occurring every time the three rollers pass throughthe interrogation points at the center of the specimen.Outside of that, the strain remains relatively constant(Fig. 4(g)), verifying that the dominant mode of deforma-tion in CBT is the superposition of bending on tension.Identified with symbols are specific instances, which cor-respond to the position of the cylinder crosshead at thebeginning and the end of the contact by the rollers.

Table 5 Initial and final (after CBT) cross-sectional geometry for samples presented in Fig. 7 to facilitate the comparison between measurements andsimulations

Specimen Initial (before testing) Final (after CBT testing to fracture)

Experiment % differencein area

Experiment Simulation % differencein area

Width[mm]

Thickness[mm]

Area[mm2]

Width[mm]

Thickness[mm]

Area[mm2]

Width[mm]

Thickness[mm]

Area[mm2]

AA6022-T4 3.5,1.2 mm/s

11.74 1.030 12.092 0.028 10.80 0.817 8.833 10.22 0.860 8.789 0.005

DP 980 (TD) 2.5,1.2 mm/s

11.77 1.01 11.887 0.011 10.38 0.812 8.436 9.88 0.837 8.269 0.019

DP 980 (TD) 3.5,1.2 mm/s

11.77 1.01 11.887 0.011 10.4 0.787 8.188 9.99 0.790 7.894 0.036

DP 1180 2.5,1.2 mm/s

11.77 1.048 12.334 0.047 10.62 0.874 9.284 10.18 0.869 8.846 0.047

DP 1180 3.5,1.2 mm/s

11.56 1.048 12.114 0.030 10.23 0.825 8.444 9.99 0.840 8.391 0.006

The initial width and thickness of the strip used in all simulations was 11.75 mm and 1 mm, respectively. ‘% difference in area’ is the % differencebetween measured and simulated areas

Fig. 9 (a) Sample curvatures resulting from bending to certain depths. (b) Maximum curvatures (i.e. 1R ) of the sample at the thickness midpoint as afunction of bending depth. The theoretical maximum defined by perfect wrapping (1/13.2 1/mm, where 13.2 = (25.4 + 1)/2) is shown as the dashed linefor reference

Exp Mech (2020) 60:459–473 469

The increment in strain based on the pulling velocity of1.2 mm/s is 0.0182, while that based on the pulling velocityof 1.35 mm/s is 0.0205. These estimates are in good agree-ment with the FEM predictions shown in Fig. 8. These strainsare smaller than the bending strain of the experimental set-up(i.e., for 1 mm-thick specimen and 25.4 mm dia. Rollers, thenominal bending strain, Δεb, is Δεb ¼ t

2R ¼ tDþt ¼ 0:038,

where t is the material thickness,D is the diameter of the roller,and R is the banding radius at the center of the strip. The lowerstrain per CBT conditions is a consequence of insufficientwrapping of the strip around the rollers. Not perfect wrappingduring the CBT process and the consequent increasing of thebending radius increases the tensile force. The insufficientwrapping indicates that the specimen “rides over” the leadingroller, trying to accommodate the enforced change in directionby minimal deformation (bending over a large region, i.e.,with small curvature, certainly much smaller than the inverseof the roller radius). However, the specimen is forced to wrapbetter around the center roller, which imparts a relatively largeincrease in both stretching and bending. Finally, the specimenis also riding over the trailing roller inducing a smaller strainthan at the center roller. Increasing the crosshead velocityrelative to the roller one could result in better wrapping ofthe specimen around the rollers, especially the leading one.The same of course could be accomplished by increasing the(normalized) bending depth, δ/t, which is illustrated inFig. 9(a). As the roller depth is increased this radius of bendingbecomes smaller until it reaches a critical value where thebending radius at the center of the strip is equal to 1/R. Thisis shown in Fig. 9(a), where the nodal coordinates of the FEmodel at the center of the specimen were output at differentlevels of bending. A fourth order polynomial is fit to the nodesdirectly underneath the center roller and two derivatives aretaken to calculate the curvature of the specimen. The maxi-mum curvature is directly responsible for the amount of bend-ing strain and stress present in the material and with morebending, the required tensile force is lower. With 1 mm sheetsit is predicted that our experimental setup would achieve nearperfect wrapping at a bend depth of approximately 6 mm (Fig.9(b)). The diagram can be used to estimate the portion of Δεb,given the bending depth. The value is half the curvature i.e.Δεb ¼ 1

2R. The remaining strain in an increment Δε is due totension. The analysis confirms that if there is more bendingstrain in a given deformation increment, the tensile force re-sponsible for creating the remaining (tensile) strain will besmaller.

Conclusions

In this paper, we have presented a systematic methodologybased on a combination of CBT material testing and FEM

simulations of the same test to extract the response of sheetmetals at large plastic strains. The method involvesperforming a CBT test on a custom built machine duringwhich the force vs displacement response is recorded. Thesheet is stretched to large plastic strains by CBTcircumventing any diffuse necking of the sample. The CBTtest is then simulated using the FEM, while the material flowstress response is iteratively extrapolated until the measuredand calculated force vs displacement responses match. Themethod is used to successfully identify the post-necking hard-ening behavior of an aluminum alloy, AA6022-T4, and twoDP steels, DP 980 and DP 1180. One measured load-displacement curve per material is used for the identificationof a flow stress curve, and then the flow stress curve is used topredict two additional measured load-displacement curves permaterial for verification. Good predictions testify that theCBT-FEM method can be used for inferring hardening curvesat high strain levels for sheet metals. Moreover, the predictionsunder different crosshead velocity and bending depth fromthose used for the identification of the flow curve demonstratethat the extrapolated material behavior is independent on CBTtest conditions. It is also shown that the method can predictincreasing as well as decreasing anisotropic hardening rate inthe post-necking regime.

Acknowledgements Authors are grateful for financial support to the U.S.National Science Foundation under the CAREER grant no. CMMI-1650641.

Appendix

The extrapolated portion of the curve representing the postnecking behavior is adjusted using an optimization procedure,while the portion of the curve measured in ST remains thesame. The procedure explored here for the extrapolation startswith a consideration of two curves selected to represent thesolution space, a perfect plasticity curve as the lower boundand a curve based on the final slope of the measured stress-strain curve as the upper bound (Fig. 10(a) for AA6022-T4and (c) for DP 980). Figure 10(b) and (d) show the predictedload vs displacement corresponding to the upper and lowerbound extrapolated stress-strain curves. As expected, thesecalculations over-predict and under-predict the load-displacement behavior of the materials indicating that the ex-trapolated curve for the materials are in between. Next, a set ofcurves in between the upper and lower bound curves is select-ed to simulate the resulting load vs displacement. Based on thecomparison between the calculated and measured load vs dis-placement, the extrapolated stress-strain curve can be roughlyestimated. After several subsequent trials, the procedure ar-rives at an accurate extrapolated curve capturing the measuredload vs displacement.

470 Exp Mech (2020) 60:459–473

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