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International Journal of Mechanical Sciences 42 (2000) 2369}2394

On the prediction of side-wall wrinkling insheet metal forming processes

Xi Wang, Jian Cao*

Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, USA

Received 6 January 1999; accepted 30 June 1999

Abstract

Prediction and prevention of side-wall wrinkling are extremely important in the design of tooling and

process parameters in sheet metal forming processes. The prediction methods can be broadly divided into

two categories: an analytical approach and a numerical simulation using "nite element method (FEM). In

this paper, a modi"ed energy approach utilizing energy equality and the e!ective dimensions of the region

undergoing circumferential compression is proposed based on simpli"ed #at or curved sheet models with

approximate boundary conditions. The analytical model calculates the critical buckling stress as a function

of material properties, geometry parameters and current in-plane stress ratio. Meanwhile, the sensitivities

of various input parameters and integration methods of FEM models on the prediction of wrinkling

phenomena are investigated. To validate our proposed method and to illustrate the sensitivity issue in theFEM simulation, comparisons with experimental results of the Yoshida buckling test, aluminum square cup

forming and aluminum conical cup forming are presented. The results demonstrate excellent agreements

between the proposed method and experiments. Our model provides a reliable and e!ective predictor for the

onset of side-wall wrinkling in sheet metal forming processes. 2000 Elsevier Science Ltd. All rights

reserved.

Keywords: Plastic buckling; Wrinkling; Sheet metal forming; Analytical solution; Energy method

1. Introduction

Wrinkling is usually undesired in "nal sheet metal parts for aesthetic or functional reasons. It is

unacceptable in the outer skin panels where the "nal part appearance is crucial. Wrinkling on the

*Corresponding author. Tel.: 847-467-1032; fax: 847-491-3915.

E-mail address: [email protected] (J. Cao).

0020-7403/00/$- see front matter 2000 Elsevier Science Ltd. All rights reserved.PII: S 0 0 2 0 - 7 4 0 3 ( 9 9 ) 0 0 0 7 8 - 8

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Nomenclature

external work done by membrane forces

; bending energyb width of the plate

a length of the platet thickness of the platew normal de#ectionx, y,x

Gcoordinates, i"1,2,3

E elastic modulus

Poisson's ratio

?@

stress components, ,"1, 2

?@

strain componentssGH

stress deviator, i,j"1, 2

GH

Kronecker deltaEQ,

Qsecant modulus and equivalent Poisson's ratio

?@GA

instantaneous moduli

M?@GA

instantaneous moduli for plane stress conditionE?@

stretching strains

?@

bending strainsN?@

membrane stress resultantsM

?@bending moments

b?@

curvature tensor of the middle surfaceu?

displacements in the in-plane directionsW

initial yield stress

V

, W

stress components

e!ective stress

e!ective strain

span angle along the hoop direction

inclination angler

, r

radii of the top and bottom in a tapered curved sheetr, r, cylindrical coordinates

PY

,P,F

stress components in cylindrical coordinates

AP

critical buckling stress

?NNJ applied compressive hoop stressAMKN

absolute value of the calculated compressive stressK material strength coe$cientn strain-hardening exponentA,B,C parameters in the Voce's lawe nominal strain in the Yoshida testeU

critical nominal strainaA, b

Ae!ective length and width

2370 X. Wang, J. Cao /International Journal of Mechanical Sciences 42 (2000) 2369}2394

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w

de#ection amplitudem, n wave number in the hoop direction and lateral/radial directionm

APcritical wave number in the hoop direction

rA

, rA

radii of the top and bottom at the e!ective compressive area

,RA, q parameters

IP, I

2, IF

,AG

parameters (i"1,2,3,4)

Fig. 1. Schematic of sheet metal forming process.

mating surfaces can adversely a!ect the part assembly and part functions, such as sealing andwelding. In addition, severe wrinkles may damage or even destroy dies. Therefore, the predictionand prevention of wrinkling are extremely important in sheet metal forming.

During the deep drawing process shown in Fig. 1, the sheet under the blank holder is drawn into

the deformation zone by the punch. As a result, compressive hoop stress and thus wrinkling can be

developed in the sheet metal under the holder (#

ange wrinkling) as well as those in the side-wall, aswrinkling is a phenomenon of compressive instability. The magnitude of the compressive stressnecessary to initiate the side-wall wrinkling is usually smaller than that for the #ange wrinklingsince the wall is relatively unsupported. Hence, the formation of side-wall wrinkles is relativelyeasier especially when the ratio of the unsupported dimension to sheet thickness is large. In

addition, the trim line of the part is usually located a little inside the die radius, and only thewrinkling in the frustum region appears in the "nal part. Hence, side-wall wrinkling is the problemof greater industrial importance and interest. The prediction on the initiation of#ange wrinkling

has been addressed analytically and numerically in a number of previous works [1}4]. A detailedreview can be found in Tomita [5] and Esche et al. [6].

Research e!orts on the prediction of wrinkling have been made in the past 50 years. Theanalytical solution can provide a global view in terms of the general tendency and the e!ect of

individual parameters on the onset of wrinkling and can be achieved in an almost negligiblecomputational time. However, past analytical work has been concentrated on some relativelysimple problems such as a column under axial loading, circular ring under inward tension, andannular plate under bending with a conical punch at the center, etc. Plastic bifurcation analysis isone of the most widely used analytical approaches to predict the onset of wrinkling. Hutchinson

and Neale [7] and Neale and Tug(cu [8] studied bifurcation phenomenon of doubly curved sheetmetal by adopting Donnell}Mushtari}Vlasov (DMV) shell approximations. The investigation was

X. Wang, J. Cao /International Journal of Mechanical Sciences 42 (2000) 2369}2394 2371

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applicable to the regions of the sheet which are free of any surface contact. Tug(cu [9,10] extended

their approach to the wrinkling of a #at plate with in"nite curvatures. Wang et al. [11] useda similar approach to study wall wrinkling for an anisotropic shell and applied the criterion toaxisymmetric shrink #anging. However, all the above analyses are limited to long wavelengthshallow mode and the boundary conditions or continuity condition along the edge of the regionbeing examined for wrinkling are neglected. Triantafyllidis [12] numerically studied the puckering

instability problem in the hemispherical cup test based on a proposed bifurcation criterion usingphenomenological corner theory instead of J2 theory. The e!ect of geometry and material

properties on the onset of non-axisymmetric plastic instability was also investigated. Fatnassi et al.[13] carried out theoretical investigations to predict the non-axisymmetric buckling in the throatof circular elastic}plastic tubes subjected a nosing operation along a frictionless conical die. Thebuckling point and associated modes are determined by Hill's bifurcation theory in conjunction

with a non-axisymmetric buckling mode. Other than bifurcation analysis, Zhang et al. [14] useda modi"ed adaptive dynamic relaxation approach to investigate the plastic wrinkling in the conicalcup using an axisymmetric model. This method allowed for the complete analysis, including

pre-failure deformation, the prediction of wrinkling and the post-wrinkling deformation. Neverthe-less, it is hard to apply this theoretical analysis in the 3-D sheet metal forming with complicated

geometry and boundary conditions. Energy method has been another approach to analyticallyinvestigate the buckling problem such as #ange wrinkling in Senior [15], Yu and Johnson [16],

Yossifon and Tirosh [17,18], Cao and Boyce [1], Wang and Cao [3] and Cao and Wang [4], etc.To our knowledge, there is still no attempt to use this energy method in studying the side-wallwrinkling.

Other than the analytical approach, experiments and numerical simulations have been conduc-ted to determine wrinkle formation tendencies in sheet metal forming. Cup forming tests with

various geometry are the common experiments to investigate side-wall wrinkling phenomenon.

Yoshida et al. [19] developed a simple test (Yoshida buckling test) to provide a reference of thewrinkling-resistant properties for various sheet metals. It involves the stretching of a square sheetalong one of its diagonals. Numerical and experimental investigations have been conducted tocorrelate Yoshida Test results with the material properties (Szacinski and Thomson [20], etc.).However, most of these results were focused on the wrinkling height while few were related to the

onset of the wrinkling. The onset and growth of wrinkles and the e!ects of material properties inthe Yoshida test were studied analytically and numerically in Tomita and Shindo [21]. Thewrinkling point was found by using Hill's bifurcation theory and Mindlin type plate theories in

conjunction with the "nite element approximation. Wang and Lee [22] employed a thin shellelement to study the wrinkling behavior of the Yoshida test. However, no extensive comparisons

with experimental results were given in either study.Numerical simulation using "nite element method (FEM) with either an implicit or explicit

integration method has become a prime tool to predict buckling behavior for the sheet metaloperation involving complicated geometry and boundary conditions including friction. Using animplicit method to predict wrinkling is essentially an eigenvalue approach, and it is hard to initiatewrinkles without initial imperfections, for example, a speci"c mode shape and/or material imperfec-tion, built into the original mesh. Unlike the implicit solver, the explicit method as a dynamic

approach can automatically generate deformed shapes with wrinkles due to the accumulation ofnumerical error. However, the onset and growth of the buckling obtained from the explicit code


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is sensitive to the input parameters in the FEM model, such as element type, mesh density,

simulation speed, etc. Generally, three types of elements are employed in the sheet metal formingsimulation, i.e. membrane element, continuum element and shell element. Membrane elementshave been widely used to model the forming processes, due to its simplicity and lower computationtime, especially in the inverse and optimization analysis where many iterations of forming arerequired. However, it does not include bending sti!ness, therefore, it may not be appropriate in

modeling the processes where the buckling phenomena is important unless some special treatment(such as post-processing) is given [2]. In general, the bending-dominant processes are simulated by

the continuum or shell elements. In continuum analysis, the bending e!ect can be taken intoaccount by having multiple layers through the thickness. However, this leads to extremely largecomputation time especially for three-dimensional problems. Shell elements may be considered asthe compromise between the continuum and membrane elements. It is possible to take into account

the e!ect of bending with much less computation time than continuum analysis although integra-tion in the thickness section is still needed. Therefore, using shell elements in explicit code becomesa possible approach to study the side-wall wrinkling in deep drawing processes. Nevertheless, as we

will discuss in this paper, the reliability of such an approach still must be veri"

ed due to itssensitivity to the FEM model parameters. This motivates us to develop a stress-based wrinkling

predictor such that we can take advantage of the reliability of FEM static (implicit) analysis for thecomplicated forming simulations and also overcome the numerical di$culty and sensitivities of the

FEM buckling analysis in both implicit and explicit methods.In the present paper, a modi"ed energy approach is presented to provide a stress-based criterion

for the side-wall wrinkling during deep drawing processes. Energy method is adopted for thedevelopment of the wrinkling predictor due to its simplicity and the previous success in predicting#ange wrinkling [3,4]. The e!ective dimensions over the region undergoing compressive hoop

stress are introduced as dimension parameters. Energy equalities for #at plate and curved sheet

models are established and a stress predictor is proposed in Section 2. To verify its predictivecapability, in Sections 3}5, the present approach is applied to the case studies of the Yoshidabuckling test, aluminum square cup forming, and aluminum conical cup forming. In each section,the corresponding model with appropriate boundary conditions is established; then the criticalbuckling stress is obtained as a function of material properties, geometrical dimensions, and stress

ratio; "nally, numerical predictions and comparisons with experimental results are reported. InSection 5, the investigation of wrinkling prediction for aluminum conical cup using various FEMsimulation methods is also given for comparison with our proposed approach in terms of the

reliability of the prediction.

2. Wrinkling criterion

This work aims to investigate the onset of wrinkling in the side-wall area of the forming part.Sheet metal in that region is free of any surface contact with the die/punch, other than the edgedisplacement restriction imposed by the tooling. For a forming part with complicated geometry,the side-wall can be divided into several relatively uniform sections, which can always be classi"ed

into two categories: #at plate and curved sheet. Therefore, in this paper, we will establishthe general formulations of wrinkling criterion and apply them for #at plate and curved sheet with


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uniform curvature in the hoop (compressive) direction using a modi"ed energy approach in the

case studies.In the following analysis, the pre-buckling stress state in the side-wall over the region examined

for wrinkling is assumed at membrane state, and thus, the shear strains and stresses are ignored. Allthe formulations are developed within the context of thin plate and shell theory, therefore, thethickness of the sheet and all the stress states through the thickness are assumed to be uniform

before buckling. Strains are expected to be small and the characteristic wavelength is largecompard to sheet thickness and yet small compared to the radii of the curvature of the sheet such

that the strain measures given by Donnell}Mushtari}Vlasov (DMV) approximations can beadopted. Deformation theory is employed in the analysis since proportional loading beforebuckling is assumed. Another limitation is that this approach cannot predict non-symmetric oranti-symmetric wrinkling.

Energy method has been extensively employed in Timoshenko [23] to study the elastic bucklingof thin plates and shells with various boundary conditions. In his energy approach, a de#ected formmay be assumed for the plate and the critical buckling condition can be assessed by equating the

internal energy of the buckled plate, ;

, and the work done by the in-plane membrane forces,

.If the internal energy for every possible assumed de#ection is larger than the work produced by

membrane forces, the sheet is under a stable equilibrium condition. Hence, the stability condition isexpressed as

);. (1)

To obtain the bending energy for every possible assumed de#ection, the formulation for generaldoubly curved sheet is employed as detailed in Hutchinson and Neale [7] and Neale and Tug(cu[8]. At the instant of buckling, the in-plane components of Lagrangian strain tensor

?@at

a distance x

from the middle surface of the curved sheet can be approximated by

?@"E

?@#x

?@

, (2)

where the Greek indices range from 1 to 2, E?@

and ?@

represent the stretching and bending strainwhich are given by

E?@"

(u?@#u

@?)#b

?@w,

?@"!w

?@, (3)

where a comma denotes covariant di!erentiation with respect to in-plane coordinates (x

,x

),u?

andw are the displacements in the in-plane direction (x

,x

) and the buckling de#ection normalto the middle surface of the sheet, b

?@is the curvature tensor of the middle surface in the

prebuckling state. If a 3-D constitutive law with the form of?@" M

?@GAGA

is adopted, where the

instantaneous moduli M?@GA are de"ned in the Appendix. The relationships for the membrane stressresultants are given by

N?@"

R

\R

?@

dx"t M

?@GAEQGA

(4)

and the bending moments are given by

M?@"

R

\R

?@x

dx"t

12M?@GAKGA

. (5)


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The strain energy under the assumption of DMV thin plate and shell theory can be obtained as

;"1 M?@ d?@#N?@ dE?@dS, (6)

where S is the region of the sheet middle surface over which the wrinkles occur. By assuming the

virtual displacements u?"0 and using deformation theory, Eq. (6) can be simpli"ed as

;"t

24 1

M?@GAwGAw?@

dS#t

2 1

M?@GAb?@bGAw dS. (7)

The external work done by the membrane forces acting in the middle plane is represented as

"1

2 1

(Nw#N

w

) dS. (8)

For a thin sheet, the boundary condition or continuity condition along the edges of the regionbeing examined for wrinkling strongly a!ects the critical buckling condition since the admissible

de#ection mode will be di!erent. By appropriately choosing the de#ection form to re#ect theboundary restriction and equating the energy ;", the critical buckling stress can be

calculated analytically as a function of in-plane stress ratio, material properties, and geometryparameters.

This wrinkling stress limit can be implemented into FEM simulation of complicated 3D formingprocesses to predict the onset of wrinkling in the sheet region free of any surface contact. A #owchart of the proposed method is shown in Fig. 2. During the FEM simulation, the principal stresses

at every material integration point in the region being examined for wrinkling are recorded. The

critical buckling stress, AP, is calculated based on the boundary conditions and the in-plane stressratio following the procedure presented above. The critical buckling stress, AP

, is then compared tothe actual compressive stress in the sheet,

?NNJ. If "

?NNJ" exceeds "

AP", wrinkling occurs.

Generally, the energy equality is considered over the entire region being examined and the stress"eld before wrinkling is assumed to be uniform over the entire region. However, from the FEM

simulation, it is found that the stress distribution is not uniform and the hoop stress is even notcompletely compressive in the frustum region. Considering that wrinkling is a phenomenon ofcompressive instability in the presence of excessive in-plane compression, here in this work, the

actual dimensions of the compressive area are employed to determine the critical bucklingcondition. In other words, the actual dimensions of the compressive area, instead of the dimensions

of the entire frustum region, will be implemented in the above energy integration in Eqs. (7) and (8).The importance of using this e!ective dimension of the compressive area will be demonstrated in

the case studies.In an e!ort to verify the predictive capability of this criterion, Yoshida Bucking test, an

aluminum square cup forming, and an aluminum conical cup forming will be studied following theprocedure illustrated in Fig. 2. In each case, the problem setup, the formulation of critical bucklingstress and the prediction results compared with experimental data will be given. For these

simulations, the commercial "nite element code ABAQUS with implicit integration solver isemployed unless otherwise speci"ed. The material is characterized as an elastic}plastic material


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Fig. 2. Flow chart of detecting the onset of wrinkling in FEM simulation using the proposed stress predictor.

following a hardening law "KL or Voce's law "A!B exp(!C), and Hill's (1948) yieldcriterion is employed to describe normal anisotropic behavior of the material.

3. Case study 1: Yoshida buckling test

3.1. Problem setup

In the Yoshida buckling test, a square sheet is stretched along a diagonal. As shown in Fig. 3,a standard test piece of 100 mm square and 0.7 mm thickness is speci"ed with a gripping width(CC and DD) of 40 mm at the corners and a gauge length (GG) of 75 mm. The nominal straine over GG at the onset of the wrinkling is de"ned as the critical nominal strain, e

U. The yield stress

of the material, mild steel, is 207 MPa, the strain-hardening exponent n is 0.22, and the materialstrength coe$cient K is 812 MPa.

3.2. Buckling condition

In the Yoshida test, a combination of tension in the y-direction and an uneven blank geometryresults in a compressive stress in the x-direction. Fig. 4 shows the contour of

VVin a quarter of the

deformed plate. Notice that the actual region under compression with e!ective length aA

ande!ective width b

Ais less than the total dimension of the plate. This problem is thus simpli"ed as the


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Fig. 3. Thin square plate subjected to tension in a diagonal direction (Yoshida buckling test).

Fig. 4. Contour of hoop stress in Yoshida buckling test and the de"nition of the e!ective dimensions in the analytical

model (unit : Pa).


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Fig. 5. Thin plate under in-plane biaxial loading.

buckling of a rectangular #at plate under plane stress condition (shown in Fig. 5) with an e!ectivelength a

A, width b

Aand thickness t. The plate is compressed by the force uniformly distributed

along the sides x"0 and aA, and is stretched by the force uniformly distributed along the sides

y"0 and bA.

Considering the possible failure modes, the boundary conditions of the rectangular plate modelare simply supported at the sides x"0, a

A, and clamped at the sides y"0, b

A. Therefore, the

de#ection surface of the buckled plate can be represented by the double sine wave as

w"w

2sinmx

aA 1!cos

2ny

bA, n,m"1,2,3,2, (9)

where w

is a constant representing the amplitude of the de#ection, m is the wave number in thecompressive x-direction, and n is the wave number in the lateral y-direction.

From Eq. (7), where b?@"0 for the #at plate, the strain energy of the plate is obtained as

;"t24

@A

?A

M*w*x

# M

*w*y

#2 M

*w

*x

*w

*y#4 M*w

*x*y

dx dy. (10)For such a plate undergoing the uniform stress "eld of a compressive stress,

V, in the x-direction

and a tensile stress, W

, in the lateral y-direction, the stress resultants are expressed asN"!t

V,

N"!t

W, and the work done by the membrane forces acting in the middle plane is given by

"!t

2

@A

?A

V

*w

*x

#

W

*w

*y

dxdy. (11)

By equating the energies calculated from Eqs. (10) and (11), ;", the critical value of thecompressive stress at a given

Wbecomes

AMKN"!

V"

t

12 Mm

aA#

1

3MaAm

2n

bA

#2

3( M

#2 M

)

2n

bA

#1

3aAm

2n

bAW

. (12)


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The critical buckling stress, AP

, is the smallest value among all the AMKN

obtained from Eq. (12)

when m varies and n equals to 1. The wave number mAP

corresponding to AP

is the critical wavenumber.

It can be seen that M?@GA

are functions of stress components, which indicates that Eq. (12) is animplicit function of the critical buckling stress,

AP, and numerical iteration may be needed to assure

that the calculated hoop stress, A?J

, coincides with the input hoop stress, GLNSR

, used for obtaining

the instantaneous moduli M?@GA

. We found that A?J

decreases monotonically with an increasing

GLNSR

. As illustrated in Fig. 2 and Section 2, wrinkling occurs if AP

is less than the actual

compressive stress in the sheet, ?NNJ

. Let GLNSR"

?NNJ, if we have

A?J'

?NNJ, that is,

A?J'

AP'

GLNSR, then

AP'

?NNJmust hold, and vice verse. Consequently, in the implementation

of the procedure shown in Fig. 2, we use ?NNJ(

A?Jas the criterion to detect the occurrence of

wrinkles to simplify the computation. No numerical iteration is, therefore, needed.

3.3. Prediction and comparison with experimental results

Due to the symmetries, a quarter of the plate is simulated in FEM using four-node reducedintegration shell elements (ABAQUS S4R) (see Fig. 3). The truncated edges D!D and C!C,where the stretching forces are applied, are considered as clamped without lateral displacement orrotation along the x and y axes. For every material integration point within the area undergoingcompressive hoop stress, the average stresses of the "ve integration points through the thickness

section are calculated, and the critical wrinkling stress is obtained from Eq. (12) using thee!ective dimensions and the average tensile stress of the integration point. Following the approach

introduced in Section 2, we could examine the occurrence of side-wall wrinkling. As shownin Fig. 6, at the nominal strain of 0.0042, the actual compressive stress in one integration

point within the examined region is equivalent or slightly greater than the calculated AP

, and

thus the onset of wrinkling is predicted to occur at this critical nominal strain. The experimentalresults in Gibson and Hobbs [24] where the wrinkling heights were recorded with respect to the

Fig. 6. Comparison of wrinkling point in Yoshida buckling test with experimental results in [24].


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nominal strain are also shown in Fig. 6 for comparison. The straight line representing their

experimental data was obtained from the curve "tting of several experimental measurements.Though the onset of wrinkling was not explicitly measured in the experiments, it is evident that ourprediction of the onset of wrinkling compares favorably with the tendency of the wrinklingheight curve.

From Eq. (12), it can be seen that the critical buckling stress strongly depends on material

properties and sheet thickness. It is evident that a thinner sheet provides a lower bucklingresistance. The e!ects of material properties are illustrated by their in#uence on instantaneous

moduli. Higher material strength component K or lower hardening exponent n yields higherwrinkling limit. This is consistent with the observation from the previous work in Ni andJhita [25].

4. Case study 2: square cup forming

From case study 1, it is indicated that the present approach provides a reasonably accurateprediction of the onset of plate wrinkling under uneven stretching. The e!ects of material properties

and sheet thickness have been quanti"ed. The following two more general case studies (Sections 4and 5) will be presented to further demonstrate the capability of the proposed approach in dealing

with two types of common cup forming tests.

4.1. Problem setup

In a practical forming process, a sheet of material is plastically deformed into a desired shape as

shown in Fig. 1. Besides the e!ects of material properties and sheet thickness on wrinkling as

addressed in the Yoshida test, the forming process involves more complicated issues such as frictione!ect, tooling e!ect and process e!ect. In particular, the frictional #at binder and/or the drawbeadare designed to provide adequate restraining force to prevent excessive metal draw-in andconsequently to eliminate the occurrence of wrinkling. Here, a square cup forming with a frictionalbinder is studied to examine the capability of our stress-based predictor in capturing the onset of

wrinkling.The forming geometry of the square cup is shown in Fig. 7. The circular blank of AL 2008-T4

with an initial diameter of 230.0 mm and a thickness of 1.0 mm was formed, where the radii of

punch, binder and die pro"les are 6.35 mm. The cross dimension of the punch is 114.3 mm with thecorner radii of 31.75 mm, and the cross dimension of die is 152.4 mm with the corner radii of

50.8 mm. The material properties used in the simulations are the initial yield stressW"125 MPa,the material strength coe$cient K"515 MPa, the strain hardening exponent n"0.26, and the

anisotropy parameter R"0.7075.


As shown in Fig. 7, the straight side-wall is constrained between the punch and the die and its

deformation should be compatible with that of the corner section. Fig. 8 shows the contour of hoopstress in a quarter of the deformed cup. Accordingly, considering the actual compressive area in the


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Fig. 7. Schematic of side-wall model in a square cup forming.

Fig. 8. Contour of hoop stress in square cup forming and the de"nition of the e!ective dimensions in the analytical model

(unit : Pa).

straight side-wall, the e!ective examined region can be simpli"ed as a plate clamped at four sides of

aA;bA (note that aA"a). Here, the e!ective dimension bA refers to the maximum width wherecompressive circumferential stress develops in the side-wall area. Hence, the de#ection satis"es thefollowing boundary conditions:

w"0,*w

*x"0 at x"0, a

A

w"0,*w

*y"0 at y"0, b

A(13)


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and it is assumed to be of the form

w"w

4 1!cos2mx

aA1!cos

2ny

bA, n,m"1,2,3,2 (14)

Similarly, for the assumed uniform stress "eld, by substituting the above admissible de#ection

form into Eqs. (10) and (11), the critical buckling stress can be obtained by letting ;":

AMKN"!

V"

t

12 M2m

aA# M

aA

2m

2n

bA

#2

3( M

#2 M

)

2n

bA

#aAm

n

bAW

. (15)

4.3. Comparisons and discussions

By taking advantage of symmetric geometry and boundary conditions, one-eighth of the entiresquare cup is simulated. Again, four-node shell elements with reduced integration (ABAQUS S4R)are used to model the blank. The punch, die and binder are modeled as rigid surfaces with frictionalinterface, where the friction coe$cient between the punch and the sheet is 0.30, and 0.17 for that

between the binder/die and the sheet [26]. The critical stress calculated from Eq. (15) is used topredict the onset of side-wall wrinkling based on the simulation results. A detailed procedure can

be found in Fig. 2 and Section 3.For a deep drawing process, the blank holder force plays an important role in the formability of

the blank. Fig. 9 displays the comparison of the critical forming height at various binder forces

before wrinkling occurs between our predictions and experimental results. It shows that the criticalforming height increases with the binder force. The analytical predictions from the present

approach are in excellent agreement with experimental results in Bakkenstuen [27]. Experimentalresults showed that when the binder force was beyond 115 kN, another type of failure, tearing,

would occur before the initiation of the wrinkling due to excessive stretching. Fig. 10 shows thedeformed shape of one buckled cup obtained from the experiment, which shows a full wave in the

draw-in direction and two full waves in the circumferential direction, compared with one full waveand three full waves, respectively, from our prediction.

In addition to material properties and sheet thickness, the tensile stress resulting from binder

design and friction also e!ects the wrinkling limit of the plate under biaxial loading. The critical

buckling stress curve with respect to the tensile stress is plotted in Fig. 11 for t"1 mm, a/t"48,and b/t"30. It is shown that the critical compressive stress decreases while the tensile stressincreases. This trend is consistent with the observation in Cao and Wang [4] where the tension in

the transverse direction reduces the wrinkling limit for the plate under normal constraint.

Note that the present approach is not applied to the analysis of cup forming where the length of the straight side is

small compared with the corner radius. In that case, the e!ect of the transition between the corner section and the straight

wall is signi"cant so that the assumed boundary condition will be inappropriate.


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Fig. 9. Comparison of critical forming heights in the square cup forming.

Fig. 10. Deformed shape of a buckled square cup [27].

Fig. 11. Calculated critical buckling stress AP

as a function of applied tensile stress W

for a given geometry.


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Fig. 12. E!ective compressive dimensionbA

versus forming height at various binder force cases in the square cup forming.

Fig. 13. E!ect of dimension b on the calculated critical buckling stress AP

.

From Fig. 9, it can be seen that the restraining force due to the blank holder force and frictioncan improve the wrinkling height. Apparently, the restraining force increases the tensile stress

component in the draw-in direction. Based on Fig. 11, it seems that a controversial conclusion wasdrawn, compared with the aforementioned prediction and experimental observation. However,from the FEM analysis, it is noticed that by increasing the restraining force, the e!ective width b

Ais

decreased for the square pan forming as shown in Fig. 12. The e!ect of the geometrical parameterbA

on the critical buckling stress is illustrated in Fig. 13. The critical buckling stress increases

dramatically when bA decreases. Therefore, it indicates that the decreased compressive width, notthe increased tensile stress, accounts for the improvement of buckling resistance. The discovery

leads to the basis for using the e!ective width undergoing compressive stress, bA, rather than the

total forming width in governing the onset of wrinkling.

5. Case study 3: conical cup forming

From the above case studies, it is shown that the introduction of e!ective dimensions ofcompressive dimension make it possible to take advantage of energy method and yield some simple


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Fig. 14. Schematic of curved wall model in a conical cup forming.

formulations of critical buckling stress. Furthermore, the predictions obtained from this modi"edenergy approach match experiments well. In this section, we will apply the theory to another

common type of geometry in sheet metal forming, i.e. conical cup or side-wall with curved section.In addition, FEM simulations using various integration methods and element types are investi-

gated for their predictability in assessing the side-wall wrinkling.

5.1. Problem setup

In an attempt to examine the capability of this stress-based predictor to capture wrinkling

of a curved sheet, a symmetric conical cup forming is investigated here. The circular blank ofAL5032-T4 with an initial diameter of 280 mm and a thickness of 1.0 mm is formed. The forming

geometry is shown in Fig. 14, where the radii of punch, binder and die pro"les are 5 mm.The diameters of the punch and die are 100 and 200 mm, respectively. The material constantsused in the simulations are R"0.92, and A"418.66 MPa, B"293.6 MPa, and C"7.112 in the

Voce's law.


In the conical cup forming, the curved wall (as shown in Fig. 14) is also restricted by thetooling at the bottom and top. Therefore, it is simpli"ed as a clamped curved sheet. Fig. 15displays the contour of hoop stress in the conical cup forming simulation. Considering possible

buckling mode and the actual compressive area in the side-wall, the e!ective region investigatedcan be considered as the curved sheet with the width b

Aas shown in Fig. 15. Similarly, the e!ective

dimension bA refers to the maximum width where compressive circumferential stress developsin the side-wall area. Let r

Aand r

Adenote the radii of the top and the bottom at the e!ective

compressive area corresponding to bA, and the de#ection satis"es the following boundary

conditions:

w"0, at r"rA

, rA

,

*w

*r"0 or

*w

*r"0, at r"r

A, rA

, (16)


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where r"r/sin , and is the inclination angle. In this analysis, only axisymmetric wrinkling

mode is considered. Thus, the de#ection of the plate is assumed to be of the form

w"w

2cos(m)(1!cos ), (17)

where "2

(r!rA)/RA with RA"rA!rA , r is the radius of any material point in the examinedregion, and m is the wave number along the circumferential direction.The expressions for the curved sheet are more complicated. Considering the curved sheet with an

in"nite curvature in one direction as shown in Fig. 16, a cylindrical coordinate system is adopted.With b

"1/r and b

"0, the general formulation for the strain energy of the curved sheet in

Fig. 15. Contour of hoop stress in conical cup forming and the de"nition of the e!ective dimensions in the analytical

model (unit : Pa).

Fig. 16. Schematic of a curved sheet model.


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IF"

PA

PA

1

r(1!cos ) dr, (23)

I2"

PA

PA

( MA# M

A#2 M

A#4 M

A

)rdr, (24)

with

A"

2

RA

cos

,

A"

RAr

sin !m

2r(1!cos )

#

3

t sin

1

r(1!cos ),

A"

RA

cos

RAr

sin !m

2r(1!cos ),

A"m

2r(1!cos )!

m

RAr

sin

.

(25)

The critical buckling stress, AP

, is the smallest value among all the AMKN

obtained from Eq. (21)

for various values ofm where n is taken to be 1. The corresponding wave number mAP

is the criticalwave number. From the above formulations, it can be seen that the critical buckling stress for thecurved sheet also depends on the local curvature over the region being examined for wrinkling in

addition to material properties, sheet thickness, e!ective compressive dimension and stress ratio.

5.3. Comparisons and discussions

Similarly as described in the FEM modeling of square cup forming, four-note shell elements(ABAQUS S4R) are employed to model one-quarter of the entire conical cup. In the simulation, the

friction coe$cient between the punch and the sheet is 0.30, and 0.15 for that between the binder/dieand the sheet. Fig. 17 displays the comparison of the critical forming height at various binder forcesfor the conical cup forming. The experimental data were provided by ALCOA. Excellent agreement

between the predictions from the present approach and experimental results is obtained.As brie#y discussed in the introduction, various "nite element models with di!erent integration

methods and element types have been used to study the onset and growth of wrinkling. Here, wewill establish some of the most widely used FEM models for simulating this conical cup forming

process and examine their predictions.Using the implicit method combined with four-node shell elements (ABAQUS S4R), with or

without mesh/material imperfection, wrinkling did not occur in the side-wall area for variousbinder forces, even when the punch displacement reaches 40 mm, which is much higher than theexperimental failure heights.

Using the explicit code with four-node shell elements (Belytschko-Tsai Shell) in a commercialFEM package LS-DYNA, wrinkling occurs naturally in the side-wall area from the very early


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Fig. 17. Comparison of critical forming heights in the conical cup forming.

Fig. 18. History of maximum wrinkle amplitude obtained from the explicit FEM simulations of the conical cup forming

with various mesh densities at a binder force of 100 kN.

state, and grows with the increasing forming height. The maximum wrinkling amplitude in theside-wall area as a function of the forming height (or punch displacement) is used to study the

wrinkling behavior. To determine the onset of wrinkling, a certain threshold for the tolerablewrinkling amplitude has to be speci"ed. It is obvious that the forming failure height is very sensitive

to the level of the speci"ed wrinkle amplitude as shown in Fig. 18. In the following analysis, thisthreshold is taken to be 0.05 mm. As evident in Fig. 18, the predicted failure heights are sensitive tothe mesh density. In addition, Fig. 19 illustrates this dependency for various binder forces. Usingthe same threshold, the failure heights from a "ner mesh are higher than those obtained froma coarse mesh. The existence of this sensitivity on mesh density makes FEM model with explicit

integration method and shell elements not very reliable in terms of predicting the wrinklingbehavior. The same phenomenon does not exist in our proposed method where the developed


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Fig. 19. Critical forming heights in the conical cup forming predicted from explicit FEM simulations with various mesh

densities (1}16;18, 2}24;24, 3}32;36, 4}48;48, 5}64;72).

Fig. 20. Critical forming heights in the conical cup forming predicted from the present approach (stress-based predictor

# implicit FEM simulation) with various mesh densities (1}16;18, 2}24;24, 3}32;36, 4}48;48, 5}64;72).

stress-based predictor is implemented in the implicit FEM analysis. Fig. 20 demonstrates that thepredicted failure heights converge at certain values as mesh density increases.

In addition to mesh density, the punch velocity used in the explicit models is another parameterthat needs to be treated carefully. Fig. 21 shows the dependence of wrinkle height on the punch

velocity under a binder force of 100 kN. Notice that the punch velocity is increased arti"cially to1}20 m/s from a typical velocity of 100 mm/s in the real forming process in order to let thesimulation "nish in a reasonable time. As shown in Fig. 21, the predicted di!erence of failure heightamong various punch velocities can be up to 1.5}2.0 mm.

The above illustration of excellent results obtained from our proposed model and the investiga-

tion on various FEM models demonstrated that the current stress-based criterion ensures thereliable and e$cient prediction of the onset of side-wall wrinkling.


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Fig. 21. E!ect of punch velocity in the explicit FEM simulation on the history of maximum wrinkle amplitude in the

conical cup forming.

6. Conclusions

Accurate prediction of side-wall wrinkling in sheet metal forming processes has been a challeng-

ing topic. As known, a pure analytical solution is suitable only for solving some simple geometryproblems. Our investigation in Section 5 on the capability of various "nite element models forpredicting the side-wall wrinkling in a conical cup forming demonstrates the limitations of thesemodels. It concludes that the implicit FEM model with shell elements may overwhelminglyoverpredict the failure heights and the predictions from explicit FEM models are sensitive to the

selected critical wrinkle heights, the mesh density, the punch velocity, etc.

To overcome these di$

culties, a method combining the analytical solution and"

nite elementmodeling is proposed here. The side-wall of even a complicated geometry can be characterized intoa combination of local #at sheets or curved sheets. The analytical model for the onset of thebuckling of an elastic}plastic #at/curved sheet is developed in this paper using the energy method(Section 2). The model assumes uniform stress distribution in the sheet and therefore yields an

upper-bound solution. Most importantly, the model introduces the concept of e!ective compres-sive dimensions, which are the actual areas under compression in the side-wall obtained from theFEM simulation. The signi"cance of this e!ective dimension on the calculated critical buckling

stress is illustrated in Section 4. By de"ning the appropriate boundary conditions, critical bucklingstresses can be obtained in terms of material properties, in-plane stress ratio, sheet thickness and

geometry parameters. The onset of wrinkling can be assessed by comparing this calculated criticalbuckling stress and the actual applied compressive stress obtained from FEM simulation.

The present approach and the analytical model have been examined for predicting the onset ofthe wrinkling in Yoshida buckling test (Section 3), aluminum square cup forming (Section 4), andaluminum conical cup forming (Section 5). Excellent agreements between our predictions andexperimental results in all these cases are obtained (Figs. 6, 9 and 17). Those cases demonstratedthat the present analysis provides a simple and e!ective way to predict the onset of side-wall

wrinkling. This approach can be implemented easily for analyzing complicated 3D formingproblems and takes the e!ect of friction and other process parameters into consideration implicitly


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by utilizing the stress states obtained from a complete FEM simulation. However, the model does

not address the issue of anti-symmetric wrinkling and wrinkling during unloading, which will beour next research target.

A similar approach, but a di!erent analytical model, for predicting the onset of#ange wrinkling(sheet wrinkling with normal constraint) was presented in Cao and Wang [4]. The combination ofthese two works provides a more complete picture for accurately and e!ectively predicting the

onset of wrinkling in sheet metal forming processes. For example, no #ange wrinkling is detected inthe conical cup forming (Section 5) using the analytical model, which is consistent with the

experimental observation. The reliability of these criteria provides engineers a robust tool indesigning/optimizing the tooling and forming parameters and therefore may eliminate the costlytrial-and-error approach.

Acknowledgements

The "nancial support for this work from NSF grant No. DMI-9713744 is greatly appreciated.We also would like to acknowledge the ALCOA research center for providing the experimentaldata and collaboration through the Alcoa Foundation award to JC.

Appendix

For a 3-D constitutive law with the form of R?@"

?@GARGA

, the instantaneous moduli aregiven by

?@GA"EQ

1#Q

1

2(

GIHJ#

GJHI

)#Q

1!2Q

GHIJ!

1

qsGHsIJ, (A.1)

where sGH

is the stress deviator and GH

denotes the Kronecker delta, the secant modulus EQ" /,

and the equivalent Poisson's ratio Q

is obtained from

QEQ

"

E#

1

21

EQ

!1

E. (A.2)

The parameter q is given as

q"(1#)2E

R3(E

Q!E

R)#1

2

3. (A.3)

The incremental moduli for plane stress condition become

M?@GA"

?@GA!?@GA

. (A.4)


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And Hill's (1948) anisotropic plasticity is employed here; therefore the e!ective stress for plane

stress problem can be expressed as

"#

#R(

!

)

1#R,

(A.5)

where R is the ratio of plastic strain in width direction to plastic strain through thickness ina uniaxial tensile test.

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