International Journal of Mechanical Sciences 52 (2010) 1756–1766

Contents lists available at ScienceDirect

International Journal of Mechanical Sciences

0020-74

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/ijmecsci

Characterising material and process variation effects on springbackrobustness for a semi-cylindrical sheet metal forming process

T. de Souza a,n, B.F. Rolfe b

a Institute for Technology Research and Innovation, Geelong Technology Precinct, Deakin University, Waurn Ponds, Victoria 3217, Australiab School of Engineering, Faculty of Science and Technology, Deakin University, Waurn Ponds, Victoria 3217, Australia

a r t i c l e i n f o

Article history:

Received 15 September 2009

Received in revised form

29 April 2010

Accepted 15 September 2010Available online 19 September 2010

Keywords:

Springback

Sheet metal forming

Stochastic modelling

Robustness

03/$ - see front matter Crown Copyright & 2

016/j.ijmecsci.2010.09.009

esponding author.

ail address: timothy.desouza@deakin.edu.au (

a b s t r a c t

Variation in the incoming sheet material and fluctuations in the press setup is unavoidable in many

stamping plants. The effect of these variations can have a large influence on the quality of the final

stamping, in particular, unpredictable springback of the sheet when the tooling is removed. While

stochastic simulation techniques have been developed to simulate this problem, there has been little

research that connects the influence of the noise sources to springback. This paper characterises the

effect of material and process variation on the robustness of springback for a semi-cylindrical channel

forming operation, which shares a similar cross-section profile as many automotive structural

components. The study was conducted using the specialised sheet metal forming package AutoFormTM

Sigma, for which a series of stochastic simulations were performed with each of the noise sources

incrementally introduced. The effective stress and effective strain scatter in a critical location of the part

was examined and a response window, which indicates the respective process robustness, was defined.

The incremental introduction of the noise sources allows the change in size of the stress–strain

response window to be tracked. The results showed that changes to process variation parameters, such

as BHP and friction coefficient, directly affect the strain component of the stress–strain response

window by altering the magnitude of external work applied to forming system. Material variation, on

the other hand, directly affected the stress component of the response window. A relationship between

the effective stress–strain response window and the variation in springback was also established.

Crown Copyright & 2010 Published by Elsevier Ltd. All rights reserved.

1. Introduction

A major obstacle in improving quality for any manufacturingprocess is the variation in the inputs and the changing conditionsover time. The sheet metal forming industry is plagued by this.Col [1] summarised the potential sources of scatter for a stampingprocess into the following categories:

�

Material variability: despite recent advances, material proper-ties, surface roughness and thickness vary from coil to coil andalso from head to tail of a coil. � Tooling variability: geometric changes in the tooling occur due to

wear over time. Further fluctuations in surface characteristics,clearances, tool material properties and tool temperature exist.

� Process variability: variation in blank holder pressure, punch

velocity, press stiffness, tooling location.

� Lubrication: a key parameter in a successful forming opera-

tion; that is difficult to control. Lubrication conditions can

010 Published by Elsevier Ltd. All righ

T. de Souza).

change globally and locally on the sheet due to temperature orsurface roughness.

� Random variability: arises from incorrect tool or sheet

positioning, or malfunction of mechanical devices.

These noise sources can affect the quality of the final stampedcomponent and can lead to reject parts in production if notaccounted for. Rejection criteria can include [2]:

�

splitting or fracture of the sheet within the trim line, � wrinkling or buckling (due to large compressive stresses), � loose metal (due to insufficient straining), and � unsatisfactory shape (caused by springback).

In particular, springback, the elastically driven shape changewhen the sheet is removed from the tooling, is a growing concernin the automotive industry. This problem is compounded by theintroduction of advanced high-strength steels (AHSS) in manyvehicle body structures. AHSS cause an increase in springback[3–5] and, perhaps, more concerning is the increased variation inspringback from stamping to stamping that has been noted [6–9].The increased magnitudes in springback can be compensated for

ts reserved.

Nomenclature

s effective stresse effective strains1 principal stress in the longitudinal directione1 principal strain in the longitudinal directionsy yield stressy distance from the mid-plane through the sheets

thicknessr0 initial bend radiusK strength coefficientn work hardening indexRp punch radius

Rd die radiusc die clearancet sheet thicknessw0 initial sheet strip widthm friction coefficientM bending momentBHP blank holder pressureBHF blank holder forceFEM finite element methodAHSS advanced high-strength steelDP dual phaseIQR interquartile rangeSPC statistical process control

T. de Souza, B.F. Rolfe / International Journal of Mechanical Sciences 52 (2010) 1756–1766 1757

by adjusting tool geometry and/or press settings. However,reducing the variation in the final part shape is more difficult toachieve and is often not considered during the die design phase.

Further to these individual noise sources, throughout the life of astamped part the variation observed is classified by a series of compo-nents, which are often attributed to different sources of variation andcan have differing downstream effects. From a study conducted bythe Auto/Steel Partnership [10] and work by Majeske and Hammet[11], the following components of variation were identified:

�

part-to-part variation: the amount of variation that can beexpected across consecutive parts produced by the processduring a given run. � Run-to-run variation: a measure of the repeatability of the die

setup process.

� Begin-to-end-of-run variation: is a measure of the processes

stability within a run and is often referred to as the mean-shiftwithin a run.

� Total variation: is the sum of all components of variation.

Hancock et al. [12] have shown (for a particular case study) thatpart-to-part variation accounted for 21% of the total variation,while the remaining 79% primarily included press-setup variation.Therefore, while the run-to-run and begin-to-end components ofvariation make up the majority of the total variation, quality toolsand techniques such as statistical process control (SPC) can helptrack and control these sources of variation and refinement of thepress setup can reduce these inconsistencies. Part-to-part variation,on the other hand, has received limited attention due to thedifficulties associated and the laborious activities associated withcapturing variation from stamping to stamping. Two main sourcesof part-to-part variation in a stamping process are from theincoming sheet material and from the process itself. Often materialsuppliers have argued that the process variation of a particularstamping plant is unstable, causing scatter in final stampingresults. Meanwhile, stamping engineers blame the inconsistency inthe material properties for the unpredictable behaviour, believingthat their process setup is stable. This would be a relatively easydebate to resolve, if every source of variation could be quantita-tively characterised. This difficulty was experienced by Rolfe et al.[13] when attempting to analyse part-to-part variation in anindustrial environment. Therefore, developing a better under-standing of the sources of part-to-part variation and their effecton the output response requires further investigation throughmore precise and controlled approaches. Numerical simulation is,therefore, an attractive option to further investigate this problem.

Traditionally, the finite element method (FEM) has been used toreduce costs and shorten product cycle times by allowing ‘virtual’sensitivity analyses and optimisation of the forming process to beperformed at the die-design stage [14,15].The next step, to include

stochastic techniques with the FEM, is steadily being taken withnumerous techniques being developed. A comprehensive review ofthese approaches has been covered by Chen and Koc [8] all of whichallow the process performance to be assessed. These approachesinvolve running multiple individual simulations with varying inputconditions prescribed by a specified input space. The input space isformulated by associating a probability distribution with each noiseinput and then the parameter values for each simulation is generatedusing an input sampling technique. These include Monte Carloschemes [16,17], design of experiment approaches [8,18] andLatin-hypercube sampling algorithms [19]. AutoFormTM Sigmaincorporates SPC techniques with its forming solver (AutoForm-Incremental), allowing the user to perform noise-based stochasticsimulations with relative ease. The user defines the noise parameters,assuming a normal distribution, and a Latin hypercube samplingapproach [20] is used to generate the input space for conductingmultiple simulations. After performing multiple simulations theoutput response space is generated, allowing the user to assess theprocess capability and robustness. Use of this tool is becoming morewidespread with a great deal of work being produced [21–24] withgood success. Using AutoForm Sigma, Sigvant [20] and Carleer [21],were able to successfully troubleshoot the cause for splitting duringthe production of an automotive structural component.

These stochastic FEM approaches are, therefore, an attractiveoption to investigate the effect of subtle variations in material andprocess parameters on the response of a sheet metal forming system.However, these techniques rely purely on generating an outputresponse based on the input space and then adjusting the processdependent result. However, this is done without necessarilydetermining the causes of the variation in the output, and whatdrives the variation in the output. Therefore, an understanding as tohow these subtle changes in input variables alter the response of theprocess would be beneficial in providing an intuitive approach tovisualising the process robustness, and is, therefore, the purpose ofthis study. This paper develops a better understanding of variation inthe following steps. Through the controlled introduction of materialand process noise sources, a number of stochastic simulations areperformed for the formation of semi-cylindrical dual phase channels.The springback response is captured and by analysing the stress andstrains in a particular region of the part, a robustness responsewindow is proposed for the process based on the materials flowcurve. By investigating the change in response window, as each noisesource is introduced, a relationship to the variation in springback isestablished.

2. Methodology

The FEM simulation used in this study has been validated anddescribed in previous work by the authors [25]. This previous

T. de Souza, B.F. Rolfe / International Journal of Mechanical Sciences 52 (2010) 1756–17661758

work created an experimental data set in which the noise sources(material property variation, blank geometry variation andprocess variation) were quantitatively measured to form prob-abilistic inputs for the stochastic simulation. One hundredexperimental dual phase channels were formed using an Erichsen(Model 145) laboratory press and the springback measured(to within an accuracy of 0.1%) to create the output distribution.The experimental process was replicated using AutoForm and astochastic noise analysis conducted using the Sigma module. Thestochastic FEM process was compared to the experimental dataset to assess both the accuracy and precision of the springbackprediction when susceptible to noise in the input parameters.Accuracy of the springback prediction had a slight negative bias of0.281 (3%) of the experimentally measured flange angle and theprecision (in terms of the interquartile range) also underpredicted by 0.211. The small error in the prediction by the FEMprovided confidence in the use of the tool for further investigation.The following section will provide a brief overview of the forminggeometry analysed and the noise sources that were quantitativelycaptured in the previous work [25]. The stochastic FEM procedureused to characterise the effect of the various noise sources will alsobe presented.

2.1. Semi-cylindrical channel

Specifically designed tooling was been developed to allow asimple and accurate method of measuring springback, as shownin Fig. 1. This part profile is representative of a typical cross-section found in many automotive structural rails and pillars. Dueto the strict requirements for crash and chassis performance thesecomponents are often extremely stiff, and in modern day vehiclesformed from AHSS. These factors compound tolerance concerns,not only due to increased magnitudes in springback, but also theincreased variation in the final part shape caused by uncontrol-lable levels of springback. This often causes downstreamassembly complications, and hence, this forms the justificationfor the selection of this forming geometry.

The semi-cylindrical punch radius (Rp) is 20 mm, the die radius(Rd) is 5 mm and the clearance between the die and punch, c, is2.1 mm, approximately equal to initial sheet thickness +5%.A small clearance was chosen to allow the use of the toolinggeometry as the reference of the ‘drawn’ profile. The nominalblank width was 20 mm and the channels were drawn to a depthof 40 mm. This allowed a fully formed semi-cylindrical channelwith approximately a 15 mm sidewall height after forming.

DIE

BLANK HOLDER

PUNCH R

Rd

204.20

150

c

Fig. 1. Tooling geometry used to form

The semi-cylindrical channel geometry was chosen instead ofthe commonly used Numisheet ‘93 U-channel [26], because, thetraditional flat-topped U-channel exhibits a large region wherebending–unbending occurs resulting in excessive sidewall curl.This region adds further complications to the through thicknessstress distribution, and also compromises the ability to preciselymeasure the amount of springback. Instead, the semi-cylindricalchannel primarily involves bending and stretching the sheet overa curved former. This relatively simple forming process is,therefore, a good starting point for analysing the effects of subtlevariation in material and process conditions on the sensitivity ofspringback. The springback has been measured by assessing theangle of the flange from horizontal.

2.2. Sources of variation

As this study primarily focuses on the variation experienced ata part-to-part level, variation in the incoming sheet material andprocess parameters are the dominant noise sources. Previouswork by the authors [25] quantitatively captured these noisesources in a series of experimental trials and the key details willbe summarised in the following sections. The noise sources weresegregated into three categories and discussed as follows:

i.

th

material property variation: this includes the yield stress andultimate tensile stress, which alters the flow behaviour of thematerial.

ii.

Blank geometry variation: fluctuations in the thickness andstrip width of each blank and

iii.

Process parameter variation: changes in the BHP and frictioncoefficient.

2.2.1. Material variation

To quantify the potential material variation, 36 tensile testswere performed according to the AS 1391-2005 standards. Thetensile specimens were blanked from the same coil, at differentlocations, hence only showing ‘in coil’ variation, which is suitablefor this study. Fig. 2 displays the material flow curves for each testin grey and the mean experimental flow curve in red. An enlargedview of the strain range near yield is also displayed. A relativelylarge dispersion of flow behaviour is observed with both yield andultimate tensile stresses varying by up to 100 MPa. Interestingly,the differences between each of the flow curves appear to be more

to

e semi-cylindrical channels.

0

100

200

300

400

500

600

700

800

True

Str

ess

(MPa

)

True Strain

Mean ExperimentalMean Power Law

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26

Fig. 2. Variation in the flow behaviour of the dual phase steel.

R² = 0.9377

600

610

620

630

640

650

660

670

680

460 470 480 490 500 510 520 530

UTS

(MPa

)

Yield Stress (MPa)

Fig. 3. Correlation between yield stress and ultimate tensile stress for the DP600

grade steel.

Table 1Distribution parameters for each of the noise parameters investigated.

Noisesource

Mean Std. dev. Min. Max. Range(%)

CV(%)

sy 489.5 MPa 12.6 MPa 467.7 MPa 535.0 MPa 13.7 2.57

UTS 633.0 MPa 14.4 MPa 605.4 MPa 674.6 MPa 10.9 2.27

t0 1.96 mm 0.01 mm 1.91 mm 1.98 mm 3.57 0.51

w0 19.55 mm 0.23 mm 18.94 mm 19.94 mm 5.12 1.18

BHP 21 MPa 0.42 MPa 19.7 MPa 22.3 MPa 12.38 2.00

m 0.135 0.0027 0.127 0.143 11.85 2.00

T. de Souza, B.F. Rolfe / International Journal of Mechanical Sciences 52 (2010) 1756–1766 1759

related to a vertical translation of the flow stress as opposed tosignificant changes in the steels strain hardening.

A standard power law approximation (Eq. (1)) was used tocharacterise the material behaviour of each test specimen. Thisallows distribution values to be assigned to each of the materialproperties to illustrate the main sources of scatter. The average fitfor the power law is represented in Fig. 2 via the solid black line.A good fit with the averaged experimental flow response for themajority of the strain range (strain range 0.2–0.18). The strain-hardening index, n, was fit within the strain range of 5–15%conforming to the standards in ULSAB ([27]).

s¼ Kenð1Þ

In order to accurately represent the true scatter of the DP600,correlation between the material parameters was performed toestablish if any dependence occurred. It was found that the yieldstress and the ultimate tensile stress were linearly dependent, asshown in Fig. 3. This agrees with the findings in the literature[20,28] and indicates that the strain hardening, n, remains fairlyconstant at 0.14 as determined from the tensile tests. This alsoconcurs with the observations in Fig. 2 that the variation in eachflow curve is primarily caused by a vertical translation, dictatedby a shift in yield stress. This material variation behaviour shall bedescribed by a ‘dependent’ material variation model for futurereference, whereby, the variation in flow behaviour observed isdue to the dependency of the ultimate tensile stress on the yield

stress. This is in contrast to an ‘independent’ material variationmodel, which permits independent variation in the yield stress,ultimate tensile stress and subsequently strain hardening proper-ties, giving a material variation span that diverges at high strains.

A quantitative characterisation of the spread in materialparameters was performed by determining the standard devia-tion, which assumes a normal distribution, and minimum andmaximum limits (which are represented in Table 1). Normalprobability plots were performed to ensure a normal distributionfit was acceptable but for succinctness is not represented here.Table 1 shows that the variation in material properties has a rangeof approximately 10–14%, which is relatively large for the smallsample size. Exacerbating this, is the fact that these samples aretaken from the same coil, therefore, coil-to-coil variations couldwell exceed this with the potential to be quite problematic.

2.2.2. Blank geometry variation

Fluctuations in the geometry of the blank before forming arecommon in many stamping plants, and this can often be a majorcause of variation in the final component as they may alter thelevels of strain subjected to the part. Therefore, thickness andstrip width measurements of each blank were taken beforeforming. The process involved measuring the sheet thickness andthe strip width of each blank at four equi-spaced points over thestrip and an average value determined. Thickness measurementswere made using a digital micrometer and strip width measure-ments with a digital vernier caliper. The variation in sheetthickness and strip width for each of the chosen operatingwindows were categorised by the distribution parameters asshown in Table 1. Sheet thicknesses varied by approximately 3%,while due to inconsistencies when cutting the blanks on aguillotine, strip widths varied by up to 7% in the worst case.

2.2.3. Process parameter variation

The blank holder pressure and friction conditions within thetooling are two typical process parameters susceptible toinconsistencies within a given stamping run. Variations in theBHP and friction coefficient are difficult to accurately capture and,therefore, a similar coefficient of variation (CV) for the materialproperty of 2% was assigned for these parameters, as shown inTable 1. This allowed comparisons to be made on par with theother noise sources. The variations in BHP and friction coefficientare based on a six standard deviation width and are normallydistributed.

2.3. Simulation procedure

The FEM simulation was performed using AutoForm v4.1.Initially, a sensitivity analysis of numerical parameters wasperformed to ensure an accurate, stable and converged solutionof the forming process was achieved. The numerical parametersused are described in Table 2 for all simulations. It was found thatthe springback was most sensitive to mesh density. Small changes

T. de Souza, B.F. Rolfe / International Journal of Mechanical Sciences 52 (2010) 1756–17661760

in the max-displacement (time step) had little effect on spring-back, but affected the stability of the solutions punch forcepredictions. Therefore, to ensure that numerical perturbationshad minimal effect, the adaptive meshing scheme (employed byAutoForm, to reduce simulation times) has been disabled and,therefore, a constant mesh geometry and time step size wasused for all simulations performed. To improve computationalefficiency a double symmetry model was implemented.

Four individual stochastic simulations were performed, wherethe noise sources were incrementally introduced, as shown byTable 3. First, only variation in the material’s mechanical proper-ties was permitted, assuming a stable process setup with nofluctuations in any processing parameters or blank geometry. Thesecond noise assessment added blank geometry variation, andthird, process parameter variation was included. The finalstochastic simulation included all process parameter variationand blank geometry variation, but maintained a consistentmaterial flow curve, which allows an opposing comparison withFEM run 1 to be made.

AutoForm [29] recommends 25 individual simulation beperformed per noise variable (as a minimum) to adequatelycapture a reliable response space. Therefore, as outlined in Table 3,125 simulations were performed for FEM runs 1, 2 and 4, whileFEM run 3, which contained six noise variables, required a largernumber of simulations to be run.

Post-processing of each individual simulation involved export-ing the major stress, major strain (at the mid-surface through thethickness) and springback profile along the symmetric plane of thecomponent starting at the punch head and ending in the flangeregion. The major stress and major strain profiles were analysed atthe completion of the drawing step, but prior to the springbackstep, allowing direct comparison from simulation to simulation at

Table 3Setup of the four noise assessment FEM simulations performed, indicating the

addition of appropriate noise parameters and the required number of individual

simulations performed.

FEM run Noise parameters No. of simulations performed

1 sy , UTS 125

2 sy , UTS, t0, w0. 125

3 sy , UTS, t0, w0, BHP, m. 175

4 t0, w0, BHP, m. 125

Table 2Description of numerical parameters used in the FEM simulation.

Numerical Parameter Value

Tools

Tool stiffness 100 N/mm3

Tool columns (Binder) Tool centre

Material

Flow curve Ludwik

Yielding criterion Hill (isotropic)

Bi-axial stress factor 1

Lubrication—all tools 0.135 (constant)

Element type—all steps Elastic–plastic shell

Meshing

Accuracy User defined

Radius penetration 0.04 mm

Initial number of elements 2500

Max refinement level 0

Layers 11

Time steps

All time step values 0.2 s

particular regions of the formed shape. Taking the measure afterspringback, would not allow this comparison as the final part shapewould vary. The stress and strain profile of the formed shape whilestill in the tooling directly influence the springback of the channeland were therefore analysed. Each major stress and major strainprofile was then standardised using linear interpolation to allowdirect comparison of values for specific distances along the partprofile. The punch force vs. punch displacement curve was alsoexported for each simulation to investigate if any trends wereevident when particular noise sources were introduced. Manuallyexporting these results can be very laborious. Therefore, anautomated script combining MATLAB and mouse/key strokerecording software was developed for the task.

3. Results and discussion

3.1. Stress and strain response

Fig. 4 displays the major stress and major strain profiles at thesheets mid-surface for two of the four FEM stochastic simulations.Each individual simulation is plotted and displayed by thegreyscale line graphs, forming a spread of stress and strainprofiles. The red dashed line represents the nominal simulation,where all noise sources are set at their mean values. Fig. 4(a)illustrates the major stress and major strain profile for FEM run 1,which contained only variation in the material’s mechanicalproperties. While for comparison, FEM run 4, which contained novariation in the material’s mechanical properties and onlyvariation in blank geometry and process conditions is shown inFig. 4(b). These two stochastic simulations are considered as thetwo extremes and will, therefore, be discussed here.

FEM run 1 shows a high level of variation in the stressresponse, while maintaining quite a consistent level of strain fromsimulation to simulation. FEM run 4, on the other hand, exhibits alarger variation in the level of strain and a relatively consistentstress response. This result suggests that variation in thematerial’s mechanical properties primarily affects the majorstress level of the formed part and that variation in blankgeometry and process conditions influence the level of strainexperienced during the forming process.

To illustrate the effects of the other stochastic simulations in acomparative manner the distribution of the major stress andmajor strain at a critical region, the parts sidewall, have beenexamined in Fig. 5. Due to the small numerical perturbations inthe stress and strain response, an average value has beentaken over the selected regions for each sample, to providea more stable measure for comparison. The sidewall region(x¼37–44 mm as indicated in Fig. 4) undergoes a large amount ofdeformation through elongation and is, therefore, quite a strain-sensitive region. Further to this, the amount of stretching in thisregion shows a strong correlation to springback, through thetransmittal of tension forces that occurs within the tooling. Inparticular, this tension affects the bending stress of the sheetunder-load experienced over the punch radius. This relationshiphas been examined in an analytical model developed by theauthors in a previous study [30] and was shown to influence thefinal level of springback and, therefore, provides justification forselecting to analyse the stress and strain behaviour in theseregions of the part prior to springback. The stress and straindistribution over the punch face region (x¼5–20 mm) was alsodetermined and plotted. Similar trends in the variation of stressand strain were observed for this region as for the sidewall, buthave not been displayed for succinctness.

Fig. 4 shows that the greatest spread in the major stressdistribution is caused when variation in mechanical properties of

1(M

Pa)

Nominal Simulation

1(M

Pa)

Distance Along Profile (mm) Distance Along Profile (mm)

Nominal Simulation

0

100

200

300

400

500

600

700

800

900

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

1

Nominal Simulation

0

100

200

300

400

500

600

700

800

900

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

1

Distance Along Profile (mm) Distance Along Profile (mm)

Nominal Simulation

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80

Fig. 4. Major stress and major strain profiles along the sectioned distance for (a) stochastic FEM simulation 1 and (b) stochastic FEM simulation 4 illustrating the

differences in strain and stress scatter between the simulation results.

T. de Souza, B.F. Rolfe / International Journal of Mechanical Sciences 52 (2010) 1756–1766 1761

the material are introduced. FEM runs 1, 2 and 3 shares virtuallythe same distribution in major stress as FEM run 1, as theypossess the same variation in the material’s mechanical proper-ties. FEM run 4, as discussed earlier, has no variation in materialproperties and, therefore, all simulations performed in this batchrely on the mean material flow curve, resulting in a relativelyconsistent major stress distribution. In FEM run 2 and 3, whereblank geometry and then friction and BHP variation areintroduced, a progressive widening of the distribution in majorstrain occurs as each of the noise sources are introduced.

3.2. Developing a response window

By examining the scatter of effective stress and effective strainfor each of the stochastic simulations performed, a window toillustrate the changes in robustness can be visualised in a similarmethod to that shown in Fig. 6. By relating this to the material’sflow curve we can start to visualise a simple robustness functionfor the forming process. This function consists of two variationcomponents: a distribution in strain, to establish a strain window;and a distribution in stress, which is determined from thevariation in flow behaviour over the observed strain region. Theexample shown in Fig. 6 shows a function with only strainvariation, this is unlike a sheet forming process where variationalso occurs in the material’s flow curve. This decomposes theresponse window and ultimately the final output distributioninto: the size of the strain window; and the size of the potentialstress window. Therefore, the behaviour of the material’s flow

curve has a large influence on the process’s robustness. So, in thecase for AHSS the robustness function would be more aggressive,due to the increased level of strain hardening compared totraditional HSS. This could suggest the reasoning for the increasein springback variation for channel type components formed fromAHSS. The approach used to develop this response window and ananalysis of the effects of the noise sources on this responsewindow will be discussed in the following section.

The averaged major stress and major strain values measured inthe sidewall (calculated as per Section 3.1) for each simulationhave been converted into the effective stress and effective strain,assuming plane strain deformation (which occurs in this case).This allows comparisons to be made with the materials tensileflow curve. Plotting these values against one another develops thesidewalls stress–strain response space for the stochastic analysis,where each data point represents each simulations result. Fig. 7displays the scatter in flow behaviour for the two extremestochastic FEM simulations (FEM run 1 and 4), where thevariation sources are either material or process driven.

FEM run 1, which possessed no variation in process parametersor blank geometry exhibits a small variation in strain andsubsequently a small strain window, as illustrated in Fig. 7.However, the variation caused by changes in the materials yieldstress creates a large variation in stress. Using the variation ineffective strain (De) as an input window and aligning it with thevariation in effective stress (Ds) as the output response, as shownby the dashed lines, the robustness of the system can begraphically displayed in terms of the rectangular shaded area.The shaded area represents the feasible stress and strain result in

Norm

alis

ed F

requ

ency

1 (MPa)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.1

0.2

0.3

0.4

0.5

0.6

Norm

alis

ed F

requ

ency

1

0.09 0.091 0.092 0.093 0.094 0.095

740 760 780 800 820 840

Fig. 5. Major stress (a) and major strain distributions (b) in the sidewall

(x¼40 mm) for each of the stochastic simulations performed.

OU

TPU

T

INPUT

Robust Region

Sensitive Region

Fig. 6. A simple example where the materials flow curve is used as a robustness

function.

650

670

690

710

730

750

770

790

0.104 0.105 0.106 0.107 0.108 0.109 0.11

Mean

Max

Min

Fig. 7. Construction of the stress–strain response window, which consists of a

strain and stress component, as represented for two of the noise analyses.

650

670

690

710

730

750

770

790

Mean

Max

Min

0.104 0.105 0.106 0.107 0.108 0.109 0.11

Fig. 8. Effective stress and effective strain scatter plot for each stochastic FEM run

performed, displaying the effect of the introduction of material and process noise

sources on the response window, indicated by the pale blue region.

T. de Souza, B.F. Rolfe / International Journal of Mechanical Sciences 52 (2010) 1756–17661762

the sidewall of the semi-cylindrical channels in its loaded state,that is before the tooling is removed, for that entire data set. Thestress behaviour of the sheet prior to unloading in the sidewall,which indicates the tension level the sheet is subject to, hasshown to influence the magnitude of springback in previous workby the authors [30].

FEM run 4, which possessed only process and blank geometryvariation, has a response space, which aligns itself on the meanmaterial flow curve. This is expected as no variation in thematerial’s mechanical properties exist. The process and blankgeometry variation, therefore, causes a broadening of strainwindow. Only a small amount of variation in the stress responsewas evident. The shapes of the response windows between FEMruns 1 and 4 are very different. FEM run 1 is primarily dominatedby the stress component of the response window, while FEM run4 is dominated more by the variation in strain.

Fig. 8 shows the effective stress and effective strain responsespace for all four of the stochastic simulations performed. It canbe seen that the introduction of further noise sources, in the caseof FEM runs 2 and 3, results in a widening of the strain window.This combined with the large stress window caused by thevariation in flow behaviour results in an increase in the size of theresponse space window, suggesting a reduction in stability of thestamping process. FEM run 3 (where all noise sources are present)exhibits the largest response window as indicated by the dashedblue lines that forms its boundaries in Fig. 8, and this is now muchlarger than that of FEM run 1 shown in Fig. 7. This result suggeststhat FEM run 3 would be more susceptible to excess springbackvariation. The effects of each noise sources on the stress–strainresponse window will be discussed in the following section.

3.2.1. Material property variation effects

Variation in the mechanical properties of the incoming sheetmaterial has shown to primarily affect the output space of thematerials stress behaviour. By analysing the trend for the effectivestress and effective strain response for the simulations performedin FEM run 1 (Fig. 7), a negative linear relationship between stress

FEM Run 1 - v, UTS FEM Run 2 - v, UTS , t0, w 0

FEM Run 3 - v, UTS , t 0, w0, BHP, µ FEM Run 4 - t 0, w0, BHP, µ

Punc

h Fo

rce

(kN)

Nominal Simulation

Punch Displacement (mm) Punch Displacement (mm)

Nominal Simulation

0

5

10

15

20

25

30

0

5

10

15

20

25

30

Punc

h Fo

rce

(kN)

Nominal Simulation

0

5

10

15

20

25

30

0

5

10

15

20

25

30

Punc

h Fo

rce

(kN)

Punc

h Fo

rce

(kN)

Punch Displacement (mm) Punch Displacement (mm)

Nominal Simulation

0 10 20 30 40 50 0 10 20 30 40 50

0 10 20 30 40 50 0 10 20 30 40 50

Fig. 9. The punch force response scatter for each of the four stochastic FEM simulations performed. An increased spread for the introduction of process and blank geometry

variation is observed due to its effect in altering the work input to the forming system.

0

0.1

0.2

0.3

0.4

0.5

0.6

Norm

alis

ed F

requ

ency

Punch Force (kN)20 20.5 21 21.5 22 22.5 23 23.5 24 24.5 25

Fig. 10. Punch force distributions for each of the stochastic simulations performed

taken at a punch stroke displacement of 34 mm.

T. de Souza, B.F. Rolfe / International Journal of Mechanical Sciences 52 (2010) 1756–1766 1763

and strain is evident. FEM run 1, which assumes no variation inprocess parameters, effectively limits the boundary conditions ofthe forming process, ensuring that constant restraining forces(caused by BHP, friction coefficient and also the blank geometry)occur and as a result a constant magnitude of external workis applied to the forming process for plastic deformation. Tofurther investigate and validate this explanation, the punch forceresponses for each of the four FEM runs were analysed, as shown inFig. 9. This figure displays the punch force response plotted foreach simulation performed per stochastic analysis. The dashed linerepresents the nominal simulation where all noise variables are setat their mean value. The punch force response exhibits two stages;the first, where the sheet is initially formed over the punch face(shown by the linear increase in force) and, the second, where arelatively steady punch force is recorded as the sheet is drawn infrom the binder to form the sidewall and slightly stretch the sheet.By comparing the scatter in the punch force response for eachof stochastic simulations, it can be seen that FEM run 1, whichexhibits no variation in process parameters, remains quiteconsistent in comparison to the other simulations, in whichprocess parameter variation was introduced. Fig. 10 furtherexplains this by representing the punch force distribution as ahistogram at a point during the steady-state punch force response(punch displacement of 34 mm). Here it is also shown that FEM run1 exhibits very little variance in the punch force, indicating that the

external work applied to the forming system remains relativelyconsistent, when no variation in parameters such as BHP, friction orblank geometry vary. These process parameters, therefore, directlyinfluence the external work and subsequently the strain energy of

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Nor

mal

ised

Fre

quen

cy

7.5 8 8.5 9 9.5 10 10.5

Fig. 11. Springback distributions for each of the stochastic simulations performed.

T. de Souza, B.F. Rolfe / International Journal of Mechanical Sciences 52 (2010) 1756–17661764

the forming system and can be detected by analysing the punchforce response. Doolan [32] documented a similar trend whencharacterising the changes in the punch force signature using aDOE approach for changes to process parameters.

Therefore, variation in the materials properties introduces alarge potential range of flow curves that the material to be formedcould be subject to. In this case, the changes in yield stress cause adirect vertical translation of the flow curve. This causes a largespread in the possible final stress profile of the formed parts andin this case a widening of the stress component in the stress–strain response window when performing multiple stampings.A fairly consistent strain response is maintained however, asvariation to material properties does not significantly alter theapplied forming forces of the system, as shown by the reasonablyconsistent punch force response profiles.

3.2.2. Blank geometry variation effects

Blank geometry variation was investigated in the form of thesheet thickness and the width of the strip. FEM run 2 introducesthese variations in addition to that of the material properties and,as described previously, results in an increase in the widening ofthe strain component of the response window. Unlike FEM run 1,which maintained a relatively constant punch force response forthe entire set of simulations, variation in blank geometry causesfluctuations in the punch force (Fig. 10), indicating that changes inthe external work of the system is occurring. In FEM run 2, constantBHP and friction are assumed. While the applied BHP does notchange, the blank holder force (BHF) does vary due to the changesin the cross-sectional area of the blank caused by changes in stripwidth, as described by Eq. (2). These subtle changes alter theapplied forming forces and, therefore, result in a change in externalwork from simulation to simulation within the stochastic set.

BHF¼ BHP�Area ð2Þ

The sheet thickness has a similar effect in altering the work ofthe forming system. In this case, the bending moment of the sheetunder load changes as fluctuations in sheet thickness occur.Eq. (3) [31] represents the bending moment of the sheet underload at a section, y, through the sheet thickness, highlights thisrelationship. The stress distribution through the sheet’s thicknesswhile still loaded in the tooling directly influences the amount ofelastic recovery that occurs when the tooling is removed and,subsequently, the amount of springback. In Eq. (3), r is thecurvature of the sheet and ds1=de1 is the plastic modulus of thesheet. The plastic modulus is altered by the magnitude ofstretching caused by the sheet was shown to affect springbacksignificantly in previous work conducted by the authors [30].

M¼

Z t=2

�t=2ðs1ydyÞ ¼

t3

12rds1

de1

����e1a ð3Þ

3.2.3. Process variation effects

Variation in process parameters, BHP and friction coefficient,directly influences the boundary conditions of the formingprocess, altering the magnitude of external work. Eq. (4)illustrates that changes in BHP (described by the normal force)and friction coefficient affect the restraining forces of the sheetwithin the tools. The magnitude of work required for plasticdeformation and the strain energy of the system, is thereforevaried. This was illustrated by FEM run 3, which introduced BHPand friction variation to blank geometry and material propertyvariation, by the further widening of the strain window.

Ff ¼ mN ð4Þ

By analysing the punch force response we can see that if alarge spread is present, it is most likely caused by variation in the

process or blank geometry parameters. These parameters primar-ily cause changes in the work required for plastic deformationand, subsequently, strain energy of the forming system. On theother hand, if the punch response is relatively stable, yet therequired response (in this case springback) appears inconsistent,it may mean that material variation is the primary cause. Withthis understanding and approach the punch force signature can,therefore, potentially be used as a diagnostic tool. It wouldsuggest potential causes of the scatter should the process startproducing defects. However, classifying the most influential noisesource would require further investigation and is not included inthe scope of this paper. Further work into process monitoring andcorrelating precise shifts and changes in the punch forcebehaviour to particular components of variation would beadvantageous to industry.

3.3. Springback response

Given that the primary focus of this work is to analyse the effectsof variation on the shape stability, the distribution of springbackfor each of the stochastic simulations performed is shown in Fig. 11as normalised histograms. The frequency in this case has beennormalised to allow direct comparisons to stochastic simulationsthat had a larger sample size (such as FEM run 3). Fig. 11 illustratesthat the stochastic simulations share a very similar mean responseof springback, although, the variance between them varies. FEM run4 shows a tighter distribution of springback, while FEM runs 1, 2 and3 have a broader spread. FEM run 1, which has only variation in thematerial’s mechanical properties, exhibits a similar distribution tothat of FEM runs 2 and 3, which possess significantly more noisesources. This indicates that variation in the material’s mechanicalproperties has more influence in compromising the parts shapestability. This result concurs with previous findings by the authors[30] and has also been reported by Chen et al. [9] and, as discussedpreviously, is primarily due to the changes in the stress componentof variation. Process and blank geometry variation (FEM run 4),which primarily altered the strain component of variation, changesthe magnitude of springback through the varied level of appliedtension as previously documented in the literature [33].

Relating the variation in springback to the response window iscritical in validating this new method of visualising springbackrobustness. To investigate this, a simple measure has beendeveloped to characterise the size of the response window bycalculating the area of the effective stress and effective strainspace. This is achieved by simply multiplying the width of thewindow (De) by its height (Ds, as shown by Eq. (5). A more

0

0.001

0.002

0.003

0.004

0.005

0.006

0

20

40

60

80

100

120

Range Effective StressRange Effective Strain

FEM Run 4

FEM Run 1

FEM Run 2

FEM Run 3

0.6 0.8 1 1.2 1.4 1.6

Fig. 13. The size of the stress and strain windows plotted against the variation in

springback. No particular dependence exists between either individual compo-

nents of the response window size.

T. de Souza, B.F. Rolfe / International Journal of Mechanical Sciences 52 (2010) 1756–1766 1765

elaborate approach, which determines the exact area of the stressand strain cluster, as opposed to purely a rectangular area may beof benefit in precisely capturing the actual output response.However, in this case, the rectangular area provides the largestpossible response window and is suitable for this discussion.

AResponse Window ¼ ðDeÞðDsÞ ð5Þ

Fig. 12 displays the calculated area of the response window foreach of the four stochastic simulations performed and itscorrelation to the springback variation, expressed as the totalrange of the part’s final flange angle. A relatively linear positivetrend is evident, highlighting that an increase in the size of theresponse window, correlates to an increase in the variation inspringback, confirming the appropriateness of using this simplemeasure. The correlation here is evident as the process stateremains quite constant, that is, no significant shifts in the level ofstrain occur. Significant changes in parameters such as BHP andfriction would cause a shift in the location of the stress–strainresponse window along the flow curve of the material. The plasticmodulus, which represents the bending stress distribution of thesheet under load and was shown to influence the mean spring-back response [30] and will change significantly with changes inthe location of the stress–strain response window. Therefore, thisstudy essentially masks any effects caused by significant changesin plastic modulus. A following study will investigate howchanges to the location of the stress–strain response windowand, subsequently the plastic modulus influence springbackvariation. It should be noted that in Fig. 12 the flange angleranges of FEM runs 1 and 4 are quite similar. However, as shownin the springback histograms (Fig. 11) and the effective stress andstrain scatter plots (Fig. 8), a few data points at the extremitiesexist and, therefore, perhaps the total range of springback is not avery good measure to distinguish between all material variationand simple process parameter variation. This result suggests thatfrom Fig. 12 as the response window size is reduced a larger dropin springback should occur.

Physically, the size of the response window represents thepotential output stress–strain state of the formed part. Further-more, it indicates the potential strain energy input of the formingprocess. In this case, where the stress and strain is analysed in thesidewall, it is linked with sheet tension and, therefore, a largeresponse window corresponds to a large variation in sheet tensionduring the deformation process. For example, one part may

y = 2.2756x + 0.8698R² = 0.9845

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Response Window Area (MPa)

FEM Run 1

FEM Run 2

FEM Run 3

FEM Run 4

0 0.05 0.1 0.15 0.2 0.25 0.3

Fig. 12. Relationship between the range of springback and the size of the response

space window for each of the four stochastic FEM simulations conducted.

experience a larger sheet tension than the next, and given thewell-established link between tension and springback, it thereforemakes sense that a large variation in tension force would cause alarge variance in the springback.

Further to the correlation shown between the responsewindow and springback, examining the effect of the size of thestress and strain window on the variation of springback was alsoperformed. This was done to investigate if either variationcomponent had more influence. Fig. 13 shows the total range inspringback as a function of both the size of the effective strainwindow (De) and effective stress window (Ds). No particulartrend is observed here for either the strain or stress window,indicating that not one particular component is suitable fordescribing the potential variation expected in springback. FEMruns 1 and 4 are of most interest, in which the major componentof variation is quite different. That is, FEM run 1 contains a largeeffective stress variation component and FEM run 4 contains alarge effective strain component, yet they appear to have a similarrange of springback. This indicates that neither strain or stressvariation dominates the variation in springback.

4. Conclusions

The two major sources of scatter, which cause part-to-partvariation are from the material and the processing conditions ofthe forming process itself. The debate between stampingengineers and material suppliers as to which source of scatterhas the greater influence has continued for some time. Adding tothis concern is the increased variation in springback caused by theintroduction of AHSS into many stamping plants. The FEM can bean attractive solution to analyse this problem and help shed somelight on this continued debate. However, previous studies are yetto characterise the fundamental causes for springback variationcaused by fluctuations in material and process conditions. Thiswas, therefore, the primary aim of this work.

Through a study that incrementally introduced material andprocess-based noise sources in a controlled manner, the devel-opment of a new methodology to construct the process’srobustness response window has been developed. By analysingchanges in the effective stress and effective strain for a criticalregion of the stamped component, a stress–strain responsewindow has been developed and characterised for each of thenoise sources introduced. A simple metric determining the area of

T. de Souza, B.F. Rolfe / International Journal of Mechanical Sciences 52 (2010) 1756–17661766

this response window was presented and its correlation to thevariation in springback shown and discussed.

This study has highlighted a number of significant findings:

�

the material’s mechanical property variation has the primaryeffect on the stress component of the response window, � variation in parameters, which primarily influence the magni-

tude of work in the stamping system, such as BHP, friction andblank geometry, affect the strain component of the responsewindow and can be detected by scatter in punch forceresponse from stamping to stamping,

� neither the strain nor the stress component was individually

responsible for the variation in springback. However, whenanalysed together, as an area of the response window,correlation to springback variation is shown for this studywhere the location of the stress–strain response windowand, subsequently, the plastic modulus, remained relativelyconstant along the materials flow curve,

� the stress–strain response window size represents the potential

variation in applied sheet tension and, therefore, a large variancein sheet tension leads to increased springback variation and

� analysis of the springback distribution highlighted that

variation in the mechanical properties of the sheet materialhad the maximum influence.

The significance of this work provides an insightful approachto understanding how the introduction of material and process-based noise influences the robustness of a forming process. Thenew method of generating a stress–strain response windowallows the robustness function of the process to be visualised. Thisapproach, therefore, indicates a direct influence of the material’sflow behaviour to the robustness response. So, in the case forAHSS, which exhibit an increased level of strain hardeningcompared to traditional HSS, the robustness function appearsmore aggressive and increases the process sensitivity. Thissuggests the reasoning for an increase in springback variationfor channel type components formed from AHSS. This approachprovides an intuitive understanding for the causes of variation inspringback and can help aid stamping engineers develop a betterunderstanding of the sensitivity of their stamping processes.

Acknowledgements

The authors would like to acknowledge Prof. J.L. Duncan andProf. P.D. Hodgson for the support and motivation provided forthis work.

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