Formability limits by wrinkling in sheet metal forming · Formability limits by wrinkling in ... in sheet metal forming. Experimentation and finite element modelling ... onset of

Original Article

Formability limits by wrinklingin sheet metal forming

JPG Magrinho, CMA Silva, MB Silva and Paulo AF Martins

Abstract

This paper presents a new combined experimental and theoretical methodology for determining the formability limits by

wrinkling in sheet metal forming. The methodology is based on the utilization of rectangular test specimens clamped

along its narrower sides and compressed lengthwise and is aimed at replicating the physics behind the occurrence of

wrinkling in deformation regions submitted to in-plane compression along one direction. The methodology draws from a

previous development in the field of flexible roll forming, and the overall objectives are to enhance and improve its

methods and procedures and to provide a new level of understanding on the onset of wrinkling in sheet metal forming.

Experimentation and finite element modelling of cylindrical deep-drawing without blank holder combined with the

utilization of the space of effective strain vs. stress triaxiality are employed to discuss the applicability and validity of

the new proposed methodology for determining the formability limits by wrinkling.

Keywords

Sheet metal forming, wrinkling, deep-drawing, experimentation, finite element method

Date received: 2 February 2016; accepted: 13 March 2016

Introduction

The formability limits in sheet metal forming set theamount of deformation that can be attained withoutfailure by necking, fracture or wrinkling. Figure 1provides a schematic representation of these limits inthe principal strain space based on an original figuremade by Marciniak1 that was recently revised byMartins et al.,2 as the result of new phenomenologicaland experimental evidence on the formability limitsby fracture.

The onset of necking is the most widely used form-ability limit in sheet metal forming, and it is generallyplotted as a ‘V-shaped’ curve, designated as the form-ing-limit curve (FLC). The FLC indicates the amountof deformation to be imparted to sheet metal parts forwhich aesthetics difficulties and incipient fracturederived from localized zones of thinning are likely tooccur.3

The onset of fracture consists of a pair of curves(designated as the fracture loci) that intersect at theupper right-hand part of the second quadrant anddelimit the strain loading conditions above whichsheet metal parts fail by cracking opening modes I(by tension) and II (by in-plane shear) of fracturemechanics.4 The fracture loci and the circumstancesleading to fracture in sheet metal forming are compre-hensively discussed in a recent state-of-the-art reviewpublished by Silva et al.5

The methodology for determining the formabilitylimits by wrinkling is not as consolidated as thoseutilized in necking and fracture. The reason for thisis attributed to the fact that wrinkling is influenced bya significantly wider and more diverse range of par-ameters that include the mechanical properties of thematerial, the geometry of the sheet metal part, the con-tact conditions imparted by the tools and the appliedlevel of stresses and strains.

The idea behind the existence of a locus of in-planestrains delimiting the onset of wrinkling in sheet metalforming is attributed to Havranek,6 who proposed theconcept of the wrinkling-limit curve (WLC) after mea-suring the circumferential and radial strains on theunsupported region of conical cups. The WLC is situ-ated in the lower left-hand of the second quadrant ofthe principal strain space (Figure 1) but its exact loca-tion is commonly determined in a case-by-case basisdepending on the regions of the sheet metal formingprocesses to be analysed.

IDMEC, Instituto Superior Tecnico, Universidade de Lisboa, Lisboa,

Portugal

Corresponding author:

Paulo AF Martins, IDMEC, Instituto Superior Tecnico, Universidade

de Lisboa, Av. Rovisco Pais, Lisboa 1049-001, Portugal.

Email: [email protected]

Proc IMechE Part L:

J Materials: Design and Applications

0(0) 1–12

! IMechE 2016

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Abe et al.,7 for example, proposed the utilization ofa conical cup drawing test performed with a flat-bot-tomed punch having a much smaller diameter thanthat of the die opening to investigate the onset ofwrinkling at the wall and flange of circular blanks.Narayanasamy and Sowerby8 utilized the same con-ical cup drawing test with two different flat-bottomedand hemispherical-ended punches and predicted theonset of wrinkling in partially draw cups by combin-ing the theories of plasticity and structural stability.They confirmed the existence of a WLC correspond-ing to critical values d"1=d"2 of the strain loading pathfor which wrinkling occurs.

The practical applicability of WLCs is sometimesquestioned because of problems related to uniquenessand existence. In fact, different strain-paths areknown to change the position of the WLCs and modi-fications in existing strain-paths to include a compres-sive in-plane strain component may eventually giverise to wrinkling anywhere in the principal strainspace.9

However, because typical strain-paths under whichwrinkling is likely to occur in sheet metal forming donot vary significantly,10 it is relevant to carry out theexperimental characterization of the WLCs undersuch representative loading conditions, so that wrink-ling can be prevented.

Yoshida et al.11 proposed the utilization of a buck-ling test consisting of a flat square sheet clamped atopposite corners and pulled in a tensile test framealong the diagonal direction, as an attempt to proposea characterization procedure that is independent fromspecific sheet metal forming processes. The test wasemployed by various authors12 to investigate the influ-ence of mechanical properties such as the yieldstrength, the strain hardening exponent and theanisotropy coefficient on the occurrence of wrinklingbut its utilization may be limited by cracking of thespecimen prior to wrinkling.

In addition to the above mentioned difficulties, theYoshida buckling test may be further limited by thesmall levels of strain that are reached in the test,which are considerably below the typical strainvalues found in real sheet metal parts. As a result ofthis, Cao et al.13 proposed a modification of theYoshida buckling test (designated as the ‘contactbuckling test’) to include a support of the blank inorder to replicate the curvature of the sheet and thecontact with the tool, which are known to delay theoccurrence of wrinkling.

The difficulty in combining the broad and diverserange of parameters that influence the onset ofwrinkling into a universal testing methodology justi-fies the development of alternative procedures thatare valid for specific sheet metal forming processes.For example, Kim and Yang14 proposed an energy-based criterion to determine the onset of wrinklingfor various sheet metal forming processes such ascylindrical, spherical and elliptical cup deep-drawing.Sivasankaran et al.15 proposed the utilization of arti-ficial neural networks to predict the onset of wrink-ling and the position of the WLC in the principalstrain space.

This paper is focused on a new testing method-ology to determine the onset of wrinkling in sheetmetal forming that circumvents the aforementionedproblems related to the occurrence of cracking priorto wrinkling. The development has its origins in thework of Kasaei et al.16 to determine the processwindow of flexible roll forming and makes use of arectangular sheet clamped at two opposite edges andcompressed in a double-action tool frame along itslongitudinal direction.

The aim and objectives of the presentation are three-fold. First, to introduce the experimental setup of thenew testing methodology and to explain its advantagein replicating the physics behind the occurrence ofwrinkling in sheet metal forming regions that generallyaccount for in-plane compression along one direction.Second, to combine experimental and finite elementresults to provide a new level of understanding of theWLC. Finally, to compare the experimental results andobservations of wrinkling obtained from cylindricaldeep-drawing without blank holder against the WLCbuilt upon the new proposed testing methodology.This last objective is crucial to evaluate the potentialof the methodology for successfully predicting theoccurrence of wrinkling in sheet metal forming.

Experimentation

Mechanical characterization of the material

The work was carried out in aluminium AA1050sheets with various thicknesses that were annealed at345 �C during 3 h prior to testing. The mechanicalcharacterization of the material was performed bytensile testing on specimens that were cut out from

Figure 1. Schematic representation of the formability limits

by necking, fracture and wrinkling in sheet metal forming.2

2 Proc IMechE Part L: J Materials: Design and Applications 0(0)



the annealed sheets at 0� and 90� degrees with respectto the rolling direction. The values obtained for themodulus of elasticity E, the yield strength �Y, theultimate tensile strength �UTS, the elongation atbreak A and the anisotropy coefficient r are given inTable 1. The stress–strain response was approximatedby the following Ludwik–Hollomon’s equation

� ¼ 151:2 "0:3 MPað Þ ð1Þ

The results obtained for the anisotropy coefficientsr0 ffi r90 ffi 1 allow concluding that the annealed alu-minium sheets behave nearly isotropically.

Characterization of the onset of wrinkling

The characterization of the WLC was performed bymeans of rectangular test specimens clamped along itsnarrower sides and compressed lengthwise(Figure 2(a)). The utilization of compressive ratherthan tensile loading as in case of the Yoshida bucklingtest is aimed at replicating the physics behind theoccurrence of wrinkling in deformation regionsundertaking in-plane compression in one direction.This is schematically illustrated in the stress-state

analogy between the rectangular test specimen andthe edge of the flange in cylindrical deep-drawingthat is shown in Figure 2(b). The outer flange elementE plotted in dark grey is subjected to circumferentialcompressive loading and may be considered as‘clamped’ along its radial edges due to deformationconstraints along the circumferential direction.

In connection to this, it is worth noting that theutilization of alternative buckling tests combining ten-sile and compression loading will not replicate thestress-state at the edge but, somewhere, inside theflange of the cylindrical cup. Therefore, they are notsuitable to reproduce the limiting strain conditionsbecause the superposition of tensile stresses will min-imize the tendency to wrinkle.

The testing apparatus developed by the authors isschematically pictured in Figure 2(a) and consists pri-marily of two parts: the frame and the self-centringclamping system. The frame is built upon a double-action cam sliding mechanism that converts the verti-cal movement of the actuator (A) into the horizontalmovement of the holders (H) containing the left andright clamping elements towards each other. The hori-zontal movement of the holders (H) produces thecompression loads that are applied on both clampedsides w0 of the test specimens. The self-centring

Figure 2. The proposed experimental test to determine the onset of wrinkling in sheet metal forming: (a) schematic representation

of the experimental setup and (b) analogy with the stress-state at the edge of the flange in cylindrical deep-drawing.

Table 1. Summary of the mechanical properties of aluminum AA1050-O sheets.

Modulus of

elasticity E (MPa)

Yield strength

�Y (MPa)

Ultimate tensile

strength �UTS (MPa)

Elongation at

break A (%)

Anisotropy

coefficient r

72472 22.1 76.2 47 r0 ¼0.97

r90 ¼1.05

Magrinho et al. 3



clamping system is a simplified version of that used byKasaei et al.16 and consists of two clamping shoes (S)and two clamping elements (C). The clamping shoes(S) serve to attach the clamping elements (C) to theholders (H). The clamping elements (C) provide thefixed end constraints of the rectangular test specimensand ensure its self-alignment land in order to preventdeformation by pure bending.

As seen in Figure 2(a), the rectangular test speci-mens subject to in-plane compression develop plasticbuckling in the centre along the width direction. Theevolution of the in-plane strain field during the testrequired electrochemical etching a grid of interlacedcircles of 2mm initial diameter d on the surface of thesheets and measuring the lengths of the major a andminor b axes of the ellipses (located on the middle ofthe width direction) that resulted from plastic deform-ation of the original grid at different instants of time(refer also to Figure 6(b)). The selected instants oftime correspond to progressively smaller distancesbetween the two opposite holders (H) containing theleft and right clamping elements.

The in-plane strain pairs ("1, "2) resulting from theabove mentioned procedure are obtained at the gridpoints located at the outer surface of the buckled testspecimens by means of conventional circle grid ana-lysis, as follows

"1 ¼ lna

d

� �"2 ¼ ln

b

d

� �ð2Þ

The deformed length l of the specimens and theamplitude h of the plastic buckled waves(Figure 2(a)) also need to be measured for furtherplots and calculations that will be addressed later inthe presentation.

Table 2 presents a summary of the rectangular spe-cimens that were used in the characterization of theonset of wrinkling by means of the new proposedtesting methodology.

Cylindrical deep-drawing benchmark cases

Cylindrical deep-drawing without blank holder wasutilized to check the occurrence of wrinkling againstthe WLC that resulted from the experiments with rect-angular test specimens that were described in the pre-vious section (section Characterization of the onset ofwrinkling).

The deep-drawing specimens consisted of circularblanks that were cut out from the annealed sheets andelectrochemically etched with a grid of interlaced cir-cles of 2mm initial diameter d on the surface of thesheets. The corresponding in-plane strains were deter-mined by circle grid analysis following a proceduresimilar to that employed in the calculation of thestrains that are needed for the characterization ofthe WLC (refer to section ‘Characterization of theonset of wrinkling’).

The geometry of the tool and circular blanks aresummarized in Table 3.

Theoretical background

Finite element modelling

The procedure to characterize the WLC by means ofthe new proposed methodology combines experimen-tal data and finite element simulation. Finite elementanalysis was carried out with the computer programLS-Dyna, and the rectangular test specimens were dis-cretized by means of 4450 shell elements with five inte-gration points across the sheet thickness (Figure 3(a)).The rectangular test specimens were simulated as elas-tic–plastic deformable objects and their mechanicalproperties are supplied in equation (1) and Table 1.

The clamping elements were modelled as rigidobjects, and the application of compressive loadingwas accomplished by imposing a constant velocity atthe constrained regions of the specimens. The desiredamount of total displacement was accomplished bymeans of an updated Lagrangian formulation thatmade use of a succession of displacement incrementseach of one corresponding to 10� 6mm in the longi-tudinal direction. The overall central processing unit(CPU) time for a typical analysis consisting of a finiteelement model similar to that shown in Figure 3(a)was 6 h on a computer equipped with an Intel�

CoreTM i7 CPU (2.93GHz) processor and makinguse of four physical cores.

Methodology to determine the onset of wrinkling

The combined experimental and numerical method-ology to determine the onset of wrinkling is

Table 2. Rectangular test specimens that were utilized in the

characterization of the onset of wrinkling.

Thickness

t0 (mm)

Length

l0(mm)

Width

w0

(mm)

l

t

w0

0

20

20

0

(mm)

1, 2, 3, 4 20 50

35

50

100




summarized in Figure 3(b) to (d). The methodology isvalid for a specific sheet thickness and first requiresensuring compatibility between the experimental andthe numerically predicted evolutions of the compres-sion force with displacement (Figure 3(b)). Second, itis necessary to calculate the critical strain pairs("crit1 , "crit2 ) at the onset of instability for the entireset of test specimens and to plot them in the principalstrain space in order to define the WLC (Figure 3(c)).Finally, it is necessary to transform the WLC from theprincipal strain space into the space of effective strain�" vs. stress triaxiality �m= �� (Figure 3(d)).

The first procedure is necessary because the onsetof wrinkling is not always attained simultaneouslyalong the entire width w of the specimens. This isparticularly noticeable for specimens with small thick-nesses and is attributed to the clamping conditionsand to the straightness and perpendicularity toler-ances in both opposite and adjacent sides of the spe-cimens. In fact, these two issues may prevent usersfrom obtaining a perfect alignment of the specimensand may even cause spurious localized deformationsby pure bending (Figure 3(b)).

These localized deformations are observed at thebeginning of the tests performed with thinner specimensand are responsible for triggering the onset of wrinklingfor smaller values of forces and displacements thanthose obtained if the specimens were perfectly flat. Asa consequence, it is necessary to introduce a quantitydesignated as the ‘initial effective width’ we

0 to adjust theexperimental settings and observations to the finiteelement simulative conditions, as follows (Figure 3(b))

we0

w0¼

Fexpcr

Ffemcr

ð3Þ

where Fexpcr and Ffem

cr are the experimental and finiteelement predicted critical forces at the onset of plasticinstability.16

The second procedure utilizes the finite elementevolution of the principal strains with the ratio h=lof the amplitude h to the length l of the deformedtest specimens (hereafter designated as the ‘normal-ized amplitude of plastic instability waves’) to deter-mine the critical strain pairs ("crit1 , "crit2 ) at the instantof time when the critical force Fcr is attained andwrinkling starts (h4 0mm, Figure 3(c)). The criticalstrain pairs for test specimens with different initiallengths l0 are then employed to construct the WLCin the principal strain space (Figure 3(d)).

In addition to finite element estimates, it is alsoimportant to obtain experimental values of the prin-cipal strains by circle grid analysis at selected instantsof time, corresponding to progressively smaller dis-tances between the two opposite holders (H) contain-ing the left and right clamping elements. Theseexperimental values were taken after the onset ofwrinkling and are mainly utilized to double-checkthe finite element simulations and to ensure the over-all quality of the predicted WLCs.

As will be seen in the following section, the WLCscan subsequently be transformed from the principalstrain space into the space of effective strain �" vs.stress triaxiality �m= ��

Representation of the WLC in the spaceof effective strain vs. stress triaxiality

The transformation of the WLCs from the principalstrain space into the space of effective strain �" vs.stress triaxiality �m= �� (Figure 3(d)) is based on theapplication of Hill’s 48 yield plasticity criterionunder plane stress conditions �3 ¼ 0

�� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�21 þ �

22 �

2r

ð1þ rÞ�1 �2

sð4Þ

Table 3. Geometry of the tool and circular blanks that were utilized in the cylindrical deep-drawing tests.

Thickness t0 (mm) Diameter D0 (mm)

1, 2, 3, 4 91

Magrinho et al. 5



In fact, by writing the constitutive equations asso-ciated to Hill’s 48 criterion that relate the major d"1and minor d"2 in-plane strain increments with theapplied stresses as follows

d"1 ¼d �"

��

1

1þ r

� ��1 þ r �1 � �2ð Þð Þ

d"2 ¼d �"

��

1

1þ r

� ��2 þ r �2 � �1ð Þð Þ ð5Þ

where the effective strain increment d �" is obtainedfrom

d �" ¼1þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ 2rÞ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid"21 þ d"22 þ

2r

ð1þ rÞd"1d"2

sð6Þ

It is possible to establish a relation between thestress triaxiality �m= �� and the slope � ¼ d"1=d"2 cor-responding to the proportional strain loading path of

Figure 3. Summary of the methodology utilized to characterize the onset of wrinkling: (a) finite element model of the rectangular

test specimen before and after compression, (b) utilization of the initial effective width we0 to ensure compatibility between the

experimental and the finite element predicted evolution of the force-displacement curves, (c) utilization of the finite element estimates

to determine the critical strain pair at the onset of wrinkling and (d) schematic representation of the wrinkling-limit curve in the

principal strain space and in the space of effective strain vs. stress triaxiality.




any arbitrary critical strain pair lying on top of theWLCs (refer, for example, to point ‘A’ in Figure 3(d))

�m��¼

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2rp

3

ð1þ �Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2r

ð1þrÞ �þ �2

q ð7Þ

The above equation allows transforming the WLCsfrom the principal strain space into the space of effect-ive strain vs. stress triaxiality that will be later utilizedin the analysis of wrinkling in cylindrical deep-drawing.

Results and discussion

Evolution of the force with displacement

Figure 4 shows the evolution of the force with dis-placement for selected rectangular specimens takenfrom the entire set of testing conditions listed inTable 2. As seen, the force increases steeply up to apeak value corresponding to the critical instabilityforce Fcr after which starts to decrease at a muchlower rate with increasing displacement. The valueof the critical instability force Fcr and the amount ofdisplacement corresponding to the onset of plasticinstability are found to increase as the initial lengthl0 decreases (Figure 4(a)) and the initial thickness t0increases (Figure 4(b)).

The particular evolution of the force with displace-ment obtained for the test specimen with l0 ¼ 20mmand t0 ¼ 4mm (Figure 4(a)) corresponds to thicken-ing without the occurrence of plastic instability. Thismode of deformation is similar to those commonlyfound in sheet-bulk metal forming,17 where the aimis to achieve thickening and/or produce local func-tional features such as ribs and solid bosses outsidethe original thickness of the sheets.

The necessity of defining an effective initial widthwe0 to ensure compatibility between the experimental

and finite element predicted evolution of force withdisplacement is illustrated in Figure 5. As seen, it isnecessary to correct the initial width w0 by the effect-ive width we

0 given by equation (3) for the test speci-mens with lower thicknesses (Figure 5(a)), but thisnecessity disappears for the test specimens withlarger thicknesses (Figure 5(b)).

The discrepancies that are observed in Figure 5(a)for the total amount of experimental and finite elem-ent predicted displacement at the peak forces is attrib-uted to the localized deformation that is observed atthe beginning of the test (refer to section‘Methodology to determine the onset of wrinkling’)because it facilitates the occurrence of wrinkling.

Determination of the onset of wrinkling

Figure 6(a) presents the finite element evolution of thein-plane strains with the normalized amplitude h=l of

plastic instability waves obtained from rectangulartest specimens subjected to the previously describedcorrection procedure. The critical strains at theonset of plastic instability ("crit1 ¼ 0:0066, "crit2 ¼

�0:0130) are obtained from the y-intercepts of themajor and minor in-plane strains (refer to the detailin Figure 6(a)) and correspond to the instant of timewhen the critical instability force Fcr is attained andbuckling is triggered (that is, when h4 0 mm). Thesquaremarkers correspond to the control experimentalvalues obtained from circle grid analysis (Figure 6(b)).

Now, by repeating the above mentioned procedurefor the entire set of test specimens that is listed inTable 2, it is possible to determine the WLCs foreach sheet thickness t0 (Figure 7(a)). As seen in thefigure, the WLCs start presenting an initial sloped"1=d"2 ¼ �1=2 similar to that of pure in-plane com-pression but later change direction towards smallerslopes as the ratio l0=t0 of the initial length l0 to theinitial thickness t0 of the rectangular test specimensdecreases.

The change in direction of the WLCs is identifiedby means of point ‘B’ in case of the rectangular testspecimens with t0 ¼ 4 mm (Figure 7(a)) and corres-ponds to a change in material flow towards anincreasing amount of thickening before wrinkling.This change in material flow is caused by the reduc-tion of the initial length l0 available to trigger andpropagate the plastic instability waves and may pre-vent the occurrence of wrinkling in situations like thatof the rectangular test specimen with l0 ¼ 20 mm andt0 ¼ 4 mm in Figure 4(a). As mentioned in previoussection ‘Evolution of the force with displacement’, themode of deformation of this particular test specimenis typical of sheet-bulk metal forming processes and,therefore, it will not provide a critical strain pair to beincluded in the definition of the WLC correspondingto t0 ¼ 4 mm.

The representation of the WLCs in the space ofeffective strain �" vs. stress triaxiality �m= ��(Figure 7(b)) results from transformation of the resultsplotted in the principal strain space (Figure 7(a)) bymeans of the analytical procedure that was previouslydescribed in section ‘Representation of theWLC in thespace of effective strain vs. stress triaxiality’. Point ‘B’illustrates the result of the transformation between thetwo spaces.

The space of effective strain �" vs. stress triaxiality�m= �� will be utilized in the following section of thepaper to compare the experimental results and obser-vations of wrinkling obtained from cylindrical deep-drawing without blank holder against the WLC builtupon the new proposed testing methodology.

Assessment by cylindrical deep-drawing

Figure 8 shows a typical finite element model of thecylindrical deep-drawing test cases that are listed inTable 3. As in case of the rectangular test specimens,

Magrinho et al. 7



the circular blanks were discretized with 11972 shellelements with five integration points across the sheetthickness. The blanks were simulated as elastic–plasticdeformable objects and its mechanical properties aregiven in equation (1) and Table 1. The punch and diewere modelled as rigid objects and discretized bymeans of spatial triangular and quadrilateral elements.

Figure 9 shows the finite element predicted loadingpaths in the space of effective strain vs. stress triaxi-ality for circular blanks with 1, 3 and 4mm thickness.The loading paths were determined for points locatedat the edge of the flanges and include valleys and hillsin case of wrinkled cylindrical cups. The experimentalvalues are plotted as squares (solid squares for valleysand open squares for hills) for the circular blanks with3 and 4mm thickness and were obtained after trans-formation of the circle grid measurements by meansof equations (6) and (7).

As seen in Figure 9(a), the finite element evolutionof the loading path for the grid points located at theedge of the flange evolves under pure compression upto point ‘B1’ where the WLC corresponding to t0 ¼ 1mm changes direction. The cause for this change indirection was already explained in section‘Determination of the onset of wrinkling’ (refer toFigure 7) and what is now relevant to understand isif and how it will influence the loading path of theselected grid points located at the edge of the flange.

In fact, after crossing point ‘B1’, the edge of theflange is no longer capable of undertaking thickeningwithout wrinkling and, therefore, its loading pathbifurcates into the two extreme loading path condi-tions of the grid points located at the bottom of thevalleys and at the top of the hills. The bottom of thevalleys continues to be thick and this is the reasonwhy its loading paths continue to evolve under pure

Figure 4. Force displacement evolution of selected rectangular test specimens of Table 2. (a) constant initial thickness t0 ¼ 4 mm

under the influence of the different initial lengths l0, (b) constant initial length l0 ¼ 50mm under the influence of different initial

thicknesses t0.




compression. In contrast, the top of the hills starts tothin and its corresponding loading paths developtowards the right side of the space of effective strainvs. stress triaxiality. Both strain loading paths are theresult of bending caused by wrinkling during whichthe neutral plane moves towards the inside of thebending regions.

The crossing point ‘B1’ can also be understoodfrom another perspective by referring to the analogybetween the new proposed test and cylindrical deep--drawing in Figure 2(b). In fact, because ‘B1’ corres-ponds to the maximum initial length lmax

0 of arectangular test specimen that undergoes wrinklingafter previous thickening exclusively under purecompression loading, it can be utilized to estimate thetotal number of wrinkles n that will be triggered in theflange of a deep-drawing cup at the onset of wrinkling

n ¼�D0

lmax0

ð8Þ

For example, in case of a sheet with t0 ¼ 1mm, themaximum initial length lmax

0 ffi 35mm (Figure 9(a))and, therefore, the expected total number of wrinklesis 8. This value is identical to that observed in thecylindrical deep-drawing experiments.

Finally, in what concerns the remaining grid pointslocated in between the bottom of the valleys and thetop of the hills, it is worth noting that they are locatedon the loading envelope given by the dotted curvethan is included in Figure 9(a), for at a particularinstant of time. In other words, all possible valuesof stress triaxiality �m= �� are located within the twoabove-mentioned extreme loading paths.

The bifurcation of the loading path at the pointwhere the WLC experiences a change in direction isalso observed for the cylindrical deep-drawing tests per-formed with a sheet thickness t0 ¼ 3 mm (Figure 9(b)).The point is now designated as ‘B3’ and the finite elem-ent predictions are in good agreement with the experi-mental values included as square markers.

Figure 5. Experimental and finite element predicted force-displacement evolution of selected rectangular test specimens of Table 2:

(a) initial length l0 ¼ 50mm and t0 ¼ 1mm and (b) initial length l0 ¼ 20mm and initial thickness t0 ¼ 4mm.

Magrinho et al. 9



The result shown in Figure 9(c) is somewhat differ-ent from those included in Figure 9(a) and (b) becausethe working range of the effective strain is now veryclose to the threshold point ‘B4’ where the WLCchanges direction. As a result of this, the signs ofbifurcation in the loading path are unnoticeable andno significant wrinkling is observed at the edge of the

flanges. This is confirmed by observation of thephotograph of the cylindrical cup included inFigure 9(c). The overall agreement with the experi-mental values is also good.

From what was mentioned above, it may be con-cluded that besides the relevance of determining theshape of the WLC by means of the new proposed

Figure 6. In-plane strains for a test specimen with l0 ¼ 50mm and t0 ¼ 2mm: (a) Experimental control values and finite element

evolution of the in-plane strains with the normalized amplitude h=l of plastic instability waves and (b) Photograph showing a buckled

test specimen and the in-plane principal strain directions.

Figure 7. Wrinkling-limit curves (WLCs) for different sheet thicknesses: (a) Representation in the principal strain space and (b)

representation in the space of effective strain vs. stress-triaxiality.




Figure 9. Assessment of the WLCs in the space of effective strain vs. stress triaxiality by means of cylindrical deep-drawing without

blank holder using circular blanks with (a) t0 ¼ 1 mm, (b) t0 ¼ 3 mm and (c) t0 ¼ 4 mm.

Figure 8. Finite element model utilized in the numerical simulation of the cylindrical deep-drawing without blank holder.

Magrinho et al. 11



testing methodology, it is of paramount importance toidentify the critical strain pair at which the slope ofthe WLC changes towards an increasing thickeningrate before wrinkling. In fact, because the edge ofthe flange in a deep-drawing operation is not capableof changing its loading path towards an increasingrate of thickening due to the stress boundary condi-tions, it will wrinkle and the loading path will bifur-cate accordingly.

Conclusions

The new testing methodology based on the double-sided compression of a rectangular sheet along itslongitudinal direction is capable of replicating thephysics behind the occurrence of wrinkling in thesheet metal forming deformation regions submittedto in-plane compression along one direction. Theonset of wrinkling is determined by combiningcircle grid analysis and finite element modelling ofthe rectangular test specimens and is plotted as theWLC in the principal strain space and in the spaceof effective strain vs. strain triaxiality. The WLC ismade of two parts, an initial straight line and a finalcurved shape that matches the increasing rate ofthickening before wrinkling. The transition pointbetween the two parts of the WLC corresponds tothe last admissible critical strain pair that is compat-ible with the stress boundary conditions that arecommonly found at the edge of the flanges ofdeep-drawing cups. Thus, any loading path goingbeyond this transition point in a deep-drawing oper-ation performed without blank holder will give riseto wrinkling and to bifurcation of the loading pathinto the two extreme conditions that are found at thebottom of the valleys and at the top of the hills. Theinitial length of a rectangular test specimen corres-ponding to the transition point can also be utilizedto estimate the number of wrinkles that will be trig-gered in the flange of a deep-drawing cup at theonset of wrinkling.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest withrespect to the research, authorship, and/or publication ofthis article.

Funding

The author(s) disclosed receipt of the following financialsupport for the research, authorship, and/or publicationof this article: The authors would like to acknowledge the

support provided by Fundacao para a Ciencia e aTecnologia of Portugal and IDMEC under LAETA –UID/EMS/50022/2013 and PDTC/EMS-TEC/0626/2014.

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