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International Journal of Mechanical Sciences 45 (2003) 1167–1180

Analytical and numerical investigation of wrinkling fordeep-drawn anisotropic metal sheets

J.P. De Magalhaes Correia∗, G. Ferron, L.P. Moreira1

Laboratoire de Physique et M�ecanique des Mat�eriaux, UMR CNRS 7554 ISGMP, Universit�e de Metz,57045 METZ Cedex 01, France

Received 4 February 2003; accepted 2 August 2003

Abstract

An analysis of the onset of wrinkling is 4rst developed for a doubly curved, elastic–plastic shell elementsubmitted to a biaxial plane stress loading. Plastic yielding is described using a criterion recently proposed foranisotropic sheet metals. The wrinkling limit curves obtained with this analysis are compared with previousresults based on di7erent yield criteria. Finite element (FE) simulations of a deep-drawing experiment are alsoperformed using the Abaqus/Explicit code with the aim of comparing the FE results relating to the initiationof wrinkling with the predictions of the analytical model and with experiments from the literature.? 2003 Elsevier Ltd. All rights reserved.

Keywords: Wrinkling; Anisotropy; Sheet metal forming; Bifurcation analysis; Finite element modelling

1. Introduction

The occurrence of wrinkling in sheet metal-forming operations depends on the current state ofstress and local geometry developed during the process in some critical region of the workpiece.Owing to the general trend towards using metal sheets exhibiting higher yield strengths and lowerthicknesses, wrinkling is a frequent defect in sheet metal forming and becomes a major problemfor manufacturers. The prediction of this defect is, therefore, of major importance for the design ofstamping and deep-drawing processes.

∗ Corresponding author. Present address: IMFS, UMR CNRS 7507, 2 rue Boussingault, 67000 Strasbourg, France. Tel.:+33-3-90-24-49-52.

E-mail address: pedro.de-magalhaes@ipst-ulp.u-strasbg.fr (J.P. De Magalhaes Correia).1 Now at: EEIMVR, Universidade Federal Fluminense, Av. dos Trabalhadores, 420 CEP 27.255.125, Volta Redonda,

RJ, Brazil.

0020-7403/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijmecsci.2003.08.001

1168 J.P. De Magalhaes Correia et al. / International Journal of Mechanical Sciences 45 (2003) 1167–1180

A large number of wrinkling analyses are based on the functional and bifurcation criterion proposedby Hutchinson [1] for the study of plastic buckling. This criterion was 4rst applied by Hutchinson andNeale [2] in a local wrinkling analysis for a doubly curved, elastic–plastic shell element submittedto a biaxial plane stress state. Plastic yielding was described in this work by either the J2-Kow orJ2-deformation theory. In the same line of approach, Neale and Tugcu [3] proposed the concept ofthe wrinkling limit curve, which corresponds to the plot of the limit stress states in principal stressaxes. Kim and Son [4] extended the local analysis of wrinkling to materials with normal anisotropydescribed by the yield function formulated by Hosford [5]. Recently, Tugcu et al. [6] investigatedthe e7ect of planar anisotropy on the wrinkling limit curves using two anisotropic yield criteriaproposed by Barlat et al. [7] and Kara4llis and Boyce [8].

Wrinkling has also been investigated by means of 4nite element (FE) simulations. In some anal-yses, initial imperfections are introduced in the mesh of the blank to initiate the occurrence of wrin-kling [9]. However, the predictions obtained in this way are sensitive to the size of the imperfections.Another approach is based on the implementation of a wrinkle indicator in the FE calculations. Thisindicator can be obtained either from energetic considerations [10], or from the de4nition of a bi-furcation functional [11,12]. Finally, several studies [13,14] have shown the eMciency of dynamicexplicit FE codes to generate the onset and the development of wrinkling in deep-drawing processeswithout the use of an initial imperfection. Furthermore, the initiation of wrinkling, as detected in FEsimulations of the conical cup test, has been found to be in fair agreement with the predictions ofa bifurcation analysis [15].

This paper aims at analysing the inKuence of planar anisotropy on the occurrence of wrinkling.A wrinkling bifurcation analysis for a doubly curved sheet is 4rst carried out using the plane stressyield function proposed by Ferron et al. [16]. This function is referred to as Ferron’s criterion in thefollowing. A parametric study of the wrinkling limit curves is presented to examine the inKuenceof the orientation of the orthotropic axes with respect to the principal stress axes, also assumingdi7erent orientations with respect to the geometric axes of principal curvatures. FE simulationsof the deep-drawing test with a tractrix die are next performed with the dynamic explicit FE codeAbaqus/Explicit, so as to compare the numerical results and the predictions of the present bifurcationanalysis with the experiments performed by Narayanasamy and Sowerby [17].

2. Wrinkling analysis

2.1. Bifurcation criterion

The bifurcation analysis employed in this work applies to thin and shallow shells with compoundcurvatures. It is similar in its principle to those developed in several previous investigations [3,4,6].The assumptions used are brieKy recalled now. The detailed calculations can be found in Ref. [15].

First, wrinkling is assumed to occur on a sheet element, with a characteristic wrinkle wavelength� that is large compared to the sheet thickness t, although small compared to the sheet radii ofcurvature (R1; R2). The conditions of continuity along the boundary of the sheet element, as well asa possible contact with a tool on the surface of the sheet element are neglected.

The di7erent axes and orientations are de4ned in Fig. 1. The direction normal to the sheet is x3.The geometric axes of principal curvatures (k1; k2) are inclined at an angle � from the principal

J.P. De Magalhaes Correia et al. / International Journal of Mechanical Sciences 45 (2003) 1167–1180 1169

Fig. 1. Axes and angles de4ning the shell geometry and the loading axes in the analytical wrinkling analysis.

in-plane orthotropic axes (x1; x2) and the principal stress axes (s1; s2) are inclined at an angle �from (x1; x2). Finally, wrinkling is assumed to occur along the direction w1, oriented at an angle �from x1.

The onset of wrinkling is analysed using the functional proposed by Hutchinson [1] in the frame-work of the Donnell–Mushtari–Vlasov shell theory. This functional F is de4ned by

F(u i; u 3) =∫S

{t3

12LijklK ijK kl + tLijklEijEkl + Niju 3; iu 3; j

}dS; (1)

where Lijkl are the components of the tangent sti7ness matrix, Nij = −t�ij are the stress resultants,Eij = 1=2(u i; j + u j; i) + biju 3 are the incremental stretching strains, bij are the curvature componentsand K ij = −u 3; ij are the incremental bending strains associated with the incremental bifurcationdisplacement 4eld (u i; u 3) = (u 1; u 2; u 3).

The calculations relating to the functional F are carried out in (w1;w2; x3) axes. The incrementalwrinkling displacement 4eld is assumed to be de4ned in these axes by

u 1 = Bt sin[�lw1

]; u 2 = 0; u 3 = At cos

[�lw1

]; (2)

where the constants A and B are the relative displacement amplitudes of the wrinkling mode and �is a non-dimensional wave number. The constant l is de4ned by l=

√Rt, where R is taken equal to

either R1 or R2. Substituting the displacement 4eld (2) into Eq. (1) and integrating over the sheetsurface S over which wrinkling spans, functional (1) can then be written in the matrix form

F =12St3u ·M · ut ; (3)

where u=(A; B) is the amplitude displacement vector and M is a square matrix of the second order.

1170 J.P. De Magalhaes Correia et al. / International Journal of Mechanical Sciences 45 (2003) 1167–1180

Wrinkling is expected to occur when the bifurcation functional F becomes equal to zero forsome non-zero displacement 4eld, that is, when the determinant of M vanishes. The critical stressesare obtained by minimising the determinant M with respect to the wave number �. The onset ofwrinkling is thus de4ned by

detM = 0;@ detM

@�= 0: (4)

These two conditions lead to the determination of the critical principal stress, �cr1 , and of the

non-dimensional wave number �. The critical compressive stress �cr1 for local wrinkling is given

by

�cr1 =

t6R

L�11

cos2(�− �) + R� sin2(�− �)�2; (5)

where R� =�2=�1 is the principal stress ratio and L�11 is the (1; 1) component of the tangent sti7ness

tensor expressed in the wrinkling axes (w1;w2). The critical wrinkling stress is a function of thewrinkling angle �, through its explicit dependence in Eq. (5), and via the orientation dependence ofL�11. The most favourable orientation for wrinkling is then obtained numerically by minimising �cr

1with respect to the orientation �. The wavelength L of the wrinkling mode is given by

L= 2�(l�

): (6)

2.2. Constitutive equations

The anisotropic plastic behaviour is described using Ferron’s yield criterion [16]. This criterionis based on a polar-coordinate representation of the yield surfaces in principal stress plane (�1; �2),as illustrated in Fig. 2. The yield surfaces are normalised by the e7ective stress, identi4ed as theequibiaxial yield stress, and they are drawn for di7erent values of the angle � between the orthotropicaxes (x1; x2) and the principal stress axes (s1; s2). In this representation, a point on the yield surfaceis de4ned by the function g( ; �) where g is the length of the polar radius and is the associatedpolar angle measured from the 4rst bisector of the plane (�1; �2). For normal anisotropy, the functiong( ) proposed in Ref. [16] is de4ned by

(1− k)g( )−6 = (cos2 + A sin2 )3 − k cos2 (cos2 − B sin2 )2; (7)

where A; B and k are dimensionless material parameters. Drucker’s yield criterion [18] is retrievedwith A= 3 and B= 9 in (7). As with Drucker’s criterion, the yield surfaces plotted with a positivek-value present Kat portions in the regions near pure shear and plane-strain tension or compression.Hill’s quadratic yield criterion [19] is also obtained as a particular case of Eq. (7) with k = 0 andA = 1 + 2r, where r denotes the coeMcient of normal anisotropy. For planar anisotropy, the yieldfunction g( ; �) is de4ned as an extension of Eq. (7), given by

g( ; �)−m = g( )−m − 2a sin cos2n−1 cos 2�+ b sin2p cos2q 2�; (8)

where m; n; p; q are positive integers and a and b are dimensionless parameters describing the materialplanar anisotropy. Hill’s quadratic yield criterion for full orthotropy is retrieved with k = 0; m= 2and n=p= q=1. The yield surfaces obtained with Ferron’s criterion present a weaker dependenceon the loading orientation � when the exponents n and p are taken larger than unity [16].

J.P. De Magalhaes Correia et al. / International Journal of Mechanical Sciences 45 (2003) 1167–1180 1171

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

g(θ,�)

= 0° = 45° = 90°

σ σ

1 /

σ σ2 /

���

θ

Fig. 2. Schematic view of the yield loci, plotted in principal stress space (�1; �2) for di7erent values of the angle �between principal orthotropic axes and principal stress axes.

Two hardening laws between e7ective stress Q� and e7ective plastic strain Q&p are employed, i.e.,the Ludwik’s law de4ned by

Q� = �y + K Q&pN

(9)

and the Swift law de4ned by

Q� = K(&0 + Q&p)N ; (10)

where �y is the yield stress, N is the strain-hardening exponent and K and &0 are material constants,respectively.

3. Wrinkling limit curves

The results of the wrinkling analysis are presented in the form of wrinkling limit curves. Thesecurves are drawn in the axes of compressive stresses (�1; �2) along the principal stress axes (s1; s2).The whole stress range is explored, at the exclusion of the biaxial tension quadrant since; as expected,wrinkling is not predicted with the bifurcation analysis in this region. In order to outline possibledi7erences with results recently obtained by Tugcu et al. [6] with di7erent yield criteria, we chooseto present the results derived with the same local geometry as in Ref. [6]. The thickness of the shellis thus taken equal to t = 1 mm, and the local curvature is de4ned by t=R1 = 0:01 and R2=R1 = 2=3.The elastic modulus E and the Poisson’s ratio ( are taken as E = 75 000 MPa and ( = 0:33,

respectively. Yielding is de4ned by the equibiaxial yield stress �y = 253 MPa and by the yieldstress values, 263.3, 264.5 and 272:2 MPa, obtained for uniaxial tension tests at 0◦; 45◦ and 90◦,respectively, from the rolling direction (RD). For these tests, the values of the anisotropy coeMcientr are equal to: r0 = 0:43; r45 = 0:84 and r90 = 1:26 at 0◦; 45◦ and 90◦, respectively. The relation

1172 J.P. De Magalhaes Correia et al. / International Journal of Mechanical Sciences 45 (2003) 1167–1180

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

= 0° = 45° = 90°

1 /

σ

σ

σ σ2 /

�

�

�

Fig. 3. Yield loci employed in the wrinkling analysis.

between the e7ective stress Q� and the e7ective plastic strain Q&p is de4ned by Ludwik’s law (9), with�y = 253 MPa; K = 440 MPa and N = 0:3225, so as to provide a best 4t of the stress–strain lawadopted in Ref. [6].

The wrinkling limit curves are calculated with Ferron’s model, adjusted to the di7erent yieldstress values de4ned above and to the anisotropy coeMcient r45 = 0:84. The corresponding materialparameters are equal to: k = 0:2969; m= n=p= 2; q= 1; A= 2:783; B= 9:138; a=−0:1056 andb = −0:1518. The yield surfaces are presented in Fig. 3, for di7erent orientations of the principalstress axes with respect to the orthotropic axes.

The Kow theory was used to compute the wrinkling limit curves in all the simulations discussedhereafter. The results are displayed in Fig. 4 for the particular case when the principal stress axes(s1; s2) are aligned with the geometric axes of principal curvatures (k1; k2), i.e., for �= � accordingto the present notations. Di7erent orientations of the orthotropic axes (x1; x2) are assumed in thecalculations. In an earlier work [15], it has been observed that the critical stress levels increase asthe radius of curvature normal to the compressive (or more compressive) direction decreases. As aresult, the critical stress states are slightly higher when the larger compressive stress is applied alongs1 (i.e., in the sector de4ned by �1 � �2), since the radius of curvature R2 is smaller. As noted byTugcu et al. [6], the e7ect of the orientation of orthotropic axes, de4ned by the angle �, is morepronounced when both principal stresses are compressive. The comparison with Fig. 3 shows thatthe level of critical wrinkling stresses (for a given stress ratio) actually varies in the same way asthe level of yield stresses as a function of the orientation � between material axes of orthotropy andprincipal stress axes.

The wrinkling limit curves obtained when the principal stress axes (s1; s2) are inclined at 45◦ fromthe geometric axes of principal curvatures (k1; k2) are shown in Fig. 5. In this case, the curve wouldbe symmetrical around �1 = �2 axis for a planar isotropic material. Again, we can observe a directcorrespondence between the variations of the critical stress state and of the yield stress level as afunction of the orientation �, taking into account the �-values simulated in Fig. 5.

J.P. De Magalhaes Correia et al. / International Journal of Mechanical Sciences 45 (2003) 1167–1180 1173

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

= η = 0° = = -30° = = -75°

σ σ

1 /

y

σ σ2 / y

�η �η �

Fig. 4. Wrinkling limit curves obtained when the principal stress axes (s1; s2) are aligned with the geometric axes (k1; k2).The principal stress axes are inclined at an angle � from the principal orthotropic axes (x1; x2). The principal stresses arenormalised by the yield stress for uniaxial tension at 0◦ from the RD direction.

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

= 45°, η = 0° = 15°, = -30° = -30°, = -75 °

σ σ

1 /

y

σ σ2 / y

�

η �η �

Fig. 5. Wrinkling limit curves obtained when the principal stress axes (s1; s2) are inclined at 45◦ from the geometric axes(k1; k2). The principal stress axes are inclined at an angle � from the principal orthotropic axes (x1; x2). The principalstresses are normalised by the yield stress for uniaxial tension at 0◦ from the RD direction.

The comparison with the results obtained in Ref. [6] shows a good agreement, since the yieldsurfaces generated in the two studies with di7erent yield criteria are similar.

The wrinkling direction is exactly aligned along either s1 or s2 when the principal stress axes,material axes and geometric axes all coincide (Fig. 4, �=�=0). The disorientation is always smallerthan 10◦ when the principal stress axes are aligned with the geometric axes of principal curvatures

1174 J.P. De Magalhaes Correia et al. / International Journal of Mechanical Sciences 45 (2003) 1167–1180

(simulations of Fig. 4). Then, the orientation of wrinkling abruptly changes from a direction close tos1 to a direction close to s2 as the stress state varies from �1 � �2 to �1 ≺ �2. However, the abruptchange is not exactly obtained for �1=�2, because of the di7erent values of principal curvature radii.Otherwise, when the principal stress axes are inclined at 45◦ from the geometric axes (simulationsof Fig. 5) a continuous variation of the wrinkling angle is observed in the vicinity of equibiaxialcompression, as the stress state varies from �1 � �2 to �1 ≺ �2. This comes from the fact that thegeometry of the shell, in this case becomes the main factor determining the orientation of wrinkles.Thus, the wrinkling direction turns towards the geometric axis k2 as the stress state approachesequibiaxial compression since this direction o7ers the lower resistance to wrinkling (R1 � R2).The normalised wrinkle wavelength L=R2 does not signi4cantly depend on the relative orientations

of the di7erent axes. On the other hand, it strongly varies as a function of the stress state, from thevalue L=R2 ≈ 1:7 near uniaxial tension to ≈ 0:3 under equibiaxial compression.

4. Wrinkling predictions for deep-drawing through a tractrix die

The analysis performed in this section aims at comparing the results from FE simulations andthe predictions of the wrinkling model with experimental results obtained by Narayanasamy andSowerby [17] for the drawing of a sheet through a tractrix die. In this process, a circular blank issimply set in a conical die with a rounded aperture, and drawn into a cylindrical cup by the actionof a cylindrical Kat-bottomed punch. The geometry of the tools is de4ned by the following values:punch diameter, 49 mm; punch pro4le radius, 5 mm; die opening diameter, 56 mm; angle of thetractrix die, 19◦. The initial thickness of the blank is equal to t = 1:90 mm, and the initial blankdiameter )initial can vary between 80.00 and 114:90 mm.

The numerical simulations have been performed with the dynamic explicit FE code Abaqus/Explicitusing Ferron’s criterion, which was previously implemented in the VUMAT user’s subroutine of thiscode [20]. Owing to the orthotropic symmetry of the material, only one-quarter of the structure ismodelled. The FE mesh of the blank is composed of 1255 elements (4-node, reduced integration,doubly curved shell elements, called S4R in Abaqus nomenclature [21]). The tools are meshed with4-node, bilinear quadrilateral, rigid elements, called R3D4 in Abaqus. The mesh of the tools and ofthe blank is shown in Fig. 6. The contact between the blank and the tools is described by Coulomb’sfriction model, where the friction coeMcient * is taken equal to 0.15.

One of the materials studied in Ref. [17] is a cold-rolled sheet of commercially pure aluminium.The anisotropy coeMcients at 0◦; 45◦ and 90◦ from the RD direction are equal to: r0=0:17; r45=0:58and r90 = 0:46, respectively. The elastic behaviour is speci4ed in the numerical simulations by thevalues of Young’s modulus, E=69 000 MPa, and of Poisson’s ratio, (=0:3. The hardening behaviouris described by the Swift law (10) with K =127:83 MPa; &0 = 0:0003 and N =0:03, so as to 4t thevalues of yield stress, ultimate tensile strength and total elongation given in Ref. [17]. The density istaken to be equal to 2700 kg=m3. The anisotropic plastic behaviour is described with Hill’s quadraticand with Ferron’s yield criteria. In all cases, the material parameters are 4tted to the r-values. Theyare found to be equal to: k = 0; m = 2; n = p = q = 1; A = 2:16; a = −0:46 and b = −0:66 forHill’s quadratic criterion [19]. The identi4cation of material parameters adopted with Ferron’s modelassume that: k=0:2; m= n=p=2; q=1 and B=3A, in agreement with the procedure proposed inRef. [20] when the material anisotropy is only speci4ed by the r-values at 0◦; 45◦ and 90◦ from the

J.P. De Magalhaes Correia et al. / International Journal of Mechanical Sciences 45 (2003) 1167–1180 1175

Fig. 6. FE mesh used for numerical simulations of the deep-drawing through a tractrix die.

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

= 0° = 45° = 90°

σ σ

1 /

2 / σ σ

�

�

�

Fig. 7. Yield loci employed in the analysis of the deep-drawing through a tractrix die.

RD direction. The other material parameters are then equal to: A= 2:35; a=−0:20 and b=−0:50.The yield surfaces obtained with Hill’s quadratic yield criterion are somewhat unrealistic, becauseof a very strong dependence of yield stresses on the orientation � of principal stress axes. The yieldsurfaces calculated with Ferron’s model are shown in Fig. 7.An example of the deformed mesh is presented in Fig. 8, for an initial blank diameter )initial =

104:94 mm and a punch stroke equal to H = 60 mm. A well-developed inward wrinkle is observedon the wall, roughly at 45◦ from the orthotropic directions. For all the values of the initial blankdiameter, wrinkling was observed to develop in this direction, which closely corresponds to themaximum value of the anisotropy coeMcient r(�), and hence to the minimum value of the uniaxialyield stress �(�). The same observation is made in the experiments performed by Narayanasamyand Sowerby [17].

1176 J.P. De Magalhaes Correia et al. / International Journal of Mechanical Sciences 45 (2003) 1167–1180

Fig. 8. An example of deformed mesh obtained for an initial blank diameter )initial = 104:94 mm and for a punch strokeH = 60 mm.

0 15 30 45 60 75-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4H = 0.43 mmH = 12.00 mmH = 36.00 mm

radi

uscu

rren

t -

rad

ius av

erag

e (

mm

)

angle from rolling direction (degrees)80

Fig. 9. Circumferential variations of the current outer radius for an initial blank diameter )initial = 84:90 mm and fordi7erent punch strokes.

In a previous study [15], the occurrence of wrinkling was captured in numerical simulations byidentifying the bifurcation point from the recording of hoop stresses for the nodes initially situated ata given radius from the pole. In the present process, wrinkling initiates on the outer radius of the cup,and the recording should thus be taken around this radius. This procedure, which is well 4tted forplanar-isotropic materials, did not give satisfactory results here, because signi4cant circumferentialvariations are observed prior to bifurcation as a result of the strong planar anisotropy of the material.A geometric criterion was then adopted to de4ne the onset of wrinkling in the numerical simulations.This criterion is de4ned by the attainment of a critical amplitude of ±0:2 mm for the circumferentialvariations of the outer radius of the cup. An example of this procedure is shown in Fig. 9 for)initial=84:90 mm. In the early stages of the drawing process, the current outer radius presents smallcircumferential variations resulting from material planar anisotropy. For a critical punch stroke equal

J.P. De Magalhaes Correia et al. / International Journal of Mechanical Sciences 45 (2003) 1167–1180 1177

70 80 90 100 110 120 130 14040

60

80

100

120experimental results:

[17]numerical results:

Hill's quadratic criterion Ferron's model

perc

enta

ge o

f dra

w-d

efor

mat

ion

initial blank diameter (mm)

Fig. 10. Comparison between experimental and numerically determined percentages of draw deformation at wrinkling, asa function of the initial blank diameter.

to H=36 mm, the current outer radius presents a well-de4ned sinusoidal variation, with an amplitudeclose to the value prescribed to de4ne the onset of wrinkling. This procedure was repeated for thedi7erent values of the initial blank diameter )initial. The achievable percentage of draw deformation,as de4ned by Narayanasamy and Sowerby [17], was also calculated. This percentage is given by

% draw deformation=)initial − )average

)initial − 56× 100; (11)

where )initial is the initial blank diameter, )average = ()outer + 56)=2 and )outer is the outer diameterof the cup when wrinkling is detected. The value of 56 mm corresponds to the outer diameter ofthe fully drawn cup.

The percentage of draw deformation at wrinkling is plotted as a function of )initial in Fig. 10, forboth Hill’s quadratic and Ferron’s criteria. The percentage of draw deformation decreases as )initial

increases, in agreement with the experimental measurements of Narayanasamy and Sowerby [17];also plotted in this 4gure. The predictions obtained with Ferron’s model are somewhat higher thanthe predictions of Hill’s criterion, and closer to the experimental results. It should also be notedthat the criterion used by Narayanasamy and Sowerby [17] to detect the onset of wrinkling is notspeci4ed in their work.

Another comparison with the results of Narayanasamy and Sowerby [17] can be made consideringthe distributions of conventional thickness strains obtained at wrinkling along the cup pro4les at0◦; 45◦ and 90◦ from the RD direction. An illustration is presented in Fig. 11 for )initial=84:90 mmand for the critical punch stroke of 36 mm. Apart from the fact that the neck which developson the radius of the punch is more pronounced than in experiments [17], the strain distributionsare well described by the FE simulations run with Ferron’s criterion. The angular dependence ofthickness strains on the external part of the cup is weak for small values of the initial blank diameter(Fig. 11), and increases for larger )initial-values, the larger thickness strains on the external radiusbeing observed at 45◦ from the RD direction. For all )initial-values, the critical thickness strain

1178 J.P. De Magalhaes Correia et al. / International Journal of Mechanical Sciences 45 (2003) 1167–1180

0 10 20 30 40 50-20

-15

-10

-5

0

5

10

15

20

-20

-15

-10

-5

0

5

10

15

20 = 0° = 45° = 90°

conv

entio

nal t

hick

ness

str

ain

distance along the profile (mm)

ααα

Fig. 11. Distribution of conventional thickness strains along the pro4les oriented at 0◦; 45◦ and 90◦ from the RD direction.The curves are drawn at the onset of wrinkling for an initial blank diameter )initial = 84:90 mm.

0 50 100 150-50

-25

0

25

500 50 100 150

-50

-25

0

25

50

= 0° = 45°and = 90°

bifurcationanalysis

H = 36mmFEM node n° 128 (0°) node n° 1028 (40.5°)

σ rr (

MP

a)

σθθ (MPa)

��

�

Fig. 12. Evolution of the stress state for two nodes at 0◦ and 40:5◦, respectively, from the RD direction. The criticalstress states obtained with the bifurcation analysis and with the FE analysis are also indicated in this 4gure.

observed for this radius at 45◦ is close to 15%, in complete agreement with the experiments. Thepredictions obtained with Hill’s quadratic criterion, not reported in this work, display an exceedinglystrong dependence of thickness strains on orientation.

The values of compressive wrinkling stresses estimated from the FE simulations were also com-pared with the predictions of the analytical model. The critical stress values are calculated in theanalytical model for uniaxial hoop compression, using the values of curvature radii attained at wrin-kling in the FE simulations. These critical stress values are pointed by an arrow in Fig. 12, for)initial = 84:90 mm. The stress paths obtained for two nodes close to the external radius of the cup,at 0◦ and 40:5◦ from the RD direction are also drawn, even though they do not provide a clearindication of the bifurcation point, as previously mentioned. Nevertheless, the critical wrinkling stress

J.P. De Magalhaes Correia et al. / International Journal of Mechanical Sciences 45 (2003) 1167–1180 1179

predicted by the analytical model is close to the value obtained in numerical simulations for thecritical punch stroke H = 36 mm, which is the value estimated for the onset of wrinkling (Fig. 9).This agreement is repeatedly observed for all )initial-values. The prediction of the analytical wrinklingmodel and the detection of the onset of wrinkling in FE simulations are thus both in fair agreementwith experimental observations.

5. Conclusions

A bifurcation analysis for curved sheet elements submitted to a biaxial stress state has been per-formed to investigate the e7ect of planar anisotropy on the onset of wrinkling in sheet metal-formingprocesses. A parametric study has been carried out, which analyses the dependence of critical stressstates on the orientation of principal stress axes with respect to both the geometric axes of principalcurvatures and the material axes of planar anisotropy. A comparison with numerical results obtainedin Ref. [6] with di7erent yield criteria displays a good agreement, as far as similar yield surfacesare generated by the various criteria.

Experimental results obtained for the deep-drawing of a cup through a tractrix die [17] were alsoanalysed to verify the reliability of the predictions obtained with the bifurcation analysis as wellas with FE numerical simulations. Both analytical and numerical predictions are in good agreementwith experiments.

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