Elastic/Crystalline Viscoplastic Finite Element Analysis of the
Deep Drawing Process, Taking Account of Texture Structure
by Hideo Morimoto* and Eiji Nakamachi *2

Earing is a phenomenon that is characteristic of deep drawing, the most commonly

ABSTRACT used method for the press forming of thin sheet, and is caused by the anisotropy ofthe material. This gives importance to techniques for controlling the texture structure by taking
account of this anisotropy to. In this way the properties of the material are used to best advantage
and adverse effects are reduced. The means increasingly used to accomplish this is finite element
analysis.
In the present work, the authors have developed a method of elastic/crystalline viscoplastic finite
element analysis which directly applies crystalline plasticity theory to the deep drawing of aluminum alloys.
Models were constructed, and different crystal orientations were introduced to investigate therelationship between deformation and texture structure, yielding appropriate analytical results thatagreed with inferences based on tensile tests of the materials.

1.

INTRODUCTION

Deep drawing is the commonest method of press forming
thin sheets, and plays an important role in 3-dimensional
forming of sheet stock. Products that depend on deep drawing are met with in many fields, from automobile bodies tohome appliances and kitchen utensils.
The most characteristic phenomena that attend deep
drawing are wrinkling and earing. Earing is an unevenness at the upper edge of the product, and is caused byanisotropy produced by biased crystal orientation, that is
to say by the texture structure, during rolling of the material. Earing is categorized in terms of type and degree into
45° earing (a total of four protrusions occurring at an angleof 45° with respect to the rolling direction of the material)
and 0/90° earing (a total of four protrusions occurring in
the rolling direction or at an angle of 90° thereto).
In recent years progress has been made in developing
crystal control techniques that result in sheets having superior formability to suppress earing and necking. The
authors have accordingly developed a method of finite element analysis using a mathematical model based on crystal structure, and have applied it to actual drawing conditions.1),2)
Conventionally in finite element analysis, anisotropy has* Materials Research Center, Yokohama Lab., R&D Div.
* 2 Materials Strength Lab., Mechanical Engineering
Dept., Osaka Institute of Technology

been modeled as changes in a yield surface, and various
proposals have been made, most conspicuously Hill's yieldfunction,3) but these have mainly been useful when load
history was comparatively simple and the parameters ofanisotropy did not vary greatly. There are thought to be
many difficulties in the way of developing a yield surface
model that takes account of the deformation-dependenceof anisotropy.
In investigating the effect of texture structure on deep
drawing, the present work, therefore, abandons the indirect approach of determining the yield function, and instead adopts a method in which crystalline plasticity theory
is applied directly to plastic working analysis. A number ofcrystal orientations are then introduced into the analyticalmodel to achieve an elastic/crystalline viscoplastic finite
element analysis.

2.

ELASTIC/CRYSTALLINE VISCOPLASTICCONSTITUTIVE EQUATIONS

In a slip system (a), let ma be the unit vector normal to aslip plane defined by a crystal, and sa be the unit vector in
the direction of slip.
These vectors rotate with deformation. Thus if m*a and*as be the vectors after deformation, we may represent thevelocity gradient tensor L, the rate of deformation tensor dand the material spin tensor w by the equations

119

..... (1)..... (2)L=d+w
d = d * +d p, w = w * + w p.
L p = d p + w p= Σ γ a s *a m *a..... (3)awhere the superscript "*" indicates the stretch and spincomponents, the superscript "p" the slip component, and.
γ a the shear strain rate of slip system (a).
The rate deformation tensor plastic portion d p and thespin tensor plastic portion wp are shown by.
.
d p = Σ P a γ a, w p = Σ W a γ aawhere τ a is the resolved shear stress, ga the reference.
shear stress and aa the reference shear strain rate of slip
system (a), and m is a strain rate sensitivity exponent. Given
that, when shear strain is zero, ga equals the critical resolved shear stress τ0, we may, with respect to ga, use thehardening evolution equation.
.N.
g a = Σ h ab γ bHere hab is a function of γ , and we adopt the equations..... (4)awhere
2P a = s *a m *a + m *a s *a, 2W a = s *a m *a – m *a s *a..... (5)If the elastic constitutive equation is unaffected by theslip deformation of the crystal, the rate type constitutive
equation will be
^

^

a=1

R a = D e : P a + β a, β a = W a . σ – σ . W a.
The time derivative of Cauchy stress σ may then be expressed by

qc ql ql
L2 = ql qv ql
ql ql qc..... (9)In a strain rate dependent crystal slip model, the shear.
strain rate in crystal slip system (a) γ a will be determinedaby Schmid's resolved shear stress τ on the slip plane, as
shown in the equationτ a = s *a . σ . m *a = P a : σ..... (15)qv ql ql
, L3 = ql qv ql
ql ql qc

and qc represents the case in which two slip systems
are within the same plane or in the same direction, qv when
they are at right angles, and ql represents other cases. For
aluminum alloy, the values of qc, qv and ql are 1.0, 1.2 and1.3.6)

3.
.
^
σ = S – σ tr L + w . σ – σ . w..... (14)qv ql ql
1 qc qc
C = qc 1 qc , L1 = ql qc ql
qc qc 1ql ql qv..... (7)..... (8)h (γ) = h 0 n C {C (γ 0 + γ)} n–1

where:

a=1

where N is the number of slip systems and Ra is asshown in the equation..... (13)C L1 L2 L3
L1 C L3 L2
q ab =L2 L3 C L1
L3 L2 L1 C

where S* is the objective co-rotational rate of Kirchhoff
stress.
A rate type constitutive equation adopting a Jaumann^rate S obtained using material spin tensor w may be found
by
N
N
.
.
^
^
S = S * – Σ βa γ a = D e : d * – Σ R a γ ah ab = h (γ) q abwhere h0 is the initial hardening ratio, C the hardening
coefficient, n the hardening exponent and γ 0 the initial shear
strain. Further, qab is a parameter representing the magnitude of self and latent hardening, and is represented by
the equation..... (6)S* = De : d*..... (12)b=1ELASTIC/CRYSTALLINE VISCOPLASTIC
FINITE ELEMENT ANALYSIS

An elastic/crystalline viscoplastic model was constructedusing the shear strain rate-dependent constitutive equation and the n-power hardening evolution equation givenin (11) and (12). At the integration point of the 8-nodeisoparametric 3-dimensional element, a "direct-crystal"..... (10)and by a parameter determined by the change in theinternal structure of the current deformation condition.
As a constitutive equation for shear strain rate γ a, we
adopted the exponential viscoplastic constitutive equationsof Hutchinson4) and Pan-Rice.5)a.
.
γa = aa τ a
g

τa
ga

1 –1
m

..... (11)

Diameter: 80 mm

Diameter: 64 mm

Figure 1 Analytical models

120.

Furukawa Review, No. 18. 1999

model7) was adopted to allocate crystal orientations.
In the present work, the results of analysis of crystal
orientation distribution and tensile tests for pure aluminum
were compared with the results of finite element analysis
of uni-axial tension to establish parameters and determine
the properties of the material.
3.1 Materials Model
One method of analyzing problems of plastic working involving anisotropy is to find the yield function by materials
testing and then conduct the analysis on the basis of thetheory of plasticity for a continuum. If, however, anisotropy
is introduced directly in the form of a crystal orientation
distribution, the analysis will automatically take into account the development of anisotropy due to deformation.
In the present work, analysis was carried out by introducing a number of crystal orientation distributions into amaterial model described as follows.
Taking account of the drawing ratios suitable for cup deep
drawing (β=1.6-2.0), we chose two types of specimens: 1)
aluminum sheet 1 mm thick and 80 mm in diameter (β=2.0)
and 2) aluminum sheet 1 mm thick and 64 mm in diameter (β=1.6). In consideration of symmetry with respect tothe rolling direction, we used the quarter portion of thesealuminum sheets. Figure 1 shows the material models usedin the analysis.
Elements were divided so that their effective lengths werevirtually identical, with a single division in the thickness
direction, giving a total of 1200 elements in model 1 and
768 in model 2, and a total number of nodes of 2522 and
1634 respectively. In both models the rolling direction was
the y-axis and the thickness direction the z-axis. Planes
parallel to the x-axis were confined in the y-axis direction,and planes parallel to the y-axis were confined in the xaxis direction.
The various material parameters used in the analysis
were identified using the results of tensile tests using purealuminum (1050-O) as described above, and are as shownin Table 1.

Figure 2 Tool data

3.2 Tools
It was assumed that the tools used for deep drawing werethe cup punch and die shown in Figure 2.
3.3 Analytical Conditions
Analyses were carried out using punch speeds of 2 m/s inthe case of Model 1 (β=2.0) and 0.5 m/s for Model 2 (β=1.6).
To determine the influence of the texture structure of thematerial, analyses were carried out based on the following three:a) Cube type (single-crystal, (100)[001]_ orientation);b) Brass type (single-crystal, (110)[112]_orientation); andc) Copper type (single-crystal, (121)[111] orientation).
{111} pole figures for each of these texture structuresare shown in Figure 5.

Table 1 Material Parameters
Material
Young's modulus

1050-OE / GPa

70.3

ρ / MPa.(mm.s-1)-2

2.7×10-7

c11 / GPac22 / GPac44 / GPa

98.88260.66228.224

Poisson's ratio

ν

0.345

Friction coefficient

Density
Anisotropic parametersµ

0.1

Initial reference shear
stressτ0 / MPa

10.0

Initial hardening ratio

h0 / MPa

37.0

Reference shear strainrate

a / s-1

0.01

Strain rate sensitivity
exponentm

0.07

Work hardening exponent

n

0.33

.

β=1.6);
Figure 3 Deformed shape of brass type structure (βoutline presents blank before deformation

121

4.

ANALYTICAL RESULTS AND DISCUSSION

4.1

Occurrence of Earing and Thickness Strain
Distribution
Figure 3 shows the results of analysis of the deformed
shape of material during the deep drawing process formaterial of brass type texture structure with a drawing ratio b of 1.6, and Figure 4 shows the results for thickness

strain distribution for cube type texture structure (β=2.0).
Figure 5 shows the results of analyses of the thickness
strain distribution at a punch stroke of approximately 30
mm (β=2.0) and of the deformed shape at a punch strokeof approximately 28 mm (β=1.6), together with {111} pole
figures of each of the three texture structures. The crystal
orientation distribution used in analysis is as shown in Figure 5.

Figure 4 Thickness strain distribution in cube type structure (cube type, β
β=2.0)a) {111} pole figureb) Thickness strain distribution (β= 2.0, Punch stroke 30mm)c) Deformed shape (β= 1.6, Punch stroke 28mm)
Cube type

Brass type

Copper typeFigure 5 Thickness strain distribution and deformed shape

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In addition, to illustrate clearly the difference in earing(deformation at the periphery of the test piece after deep
drawing), Figure 6 shows the flange edge deformation
when β=2.0 at a punch stroke of 27 mm, with the angle
from the rolling direction on the horizontal axis and theflange radius in that direction on the vertical axis. Similarly, Figure 7 shows edge deformation when β=1.6 (noflange was formed since test piece diameter was small) ata punch stroke of 28 mm.
From Figures 6 and 7 we can see that for cube-type
texture structure, deformation is greatest for material at
about 45° from the rolling direction and is drawn into thedie cavity, whereas in the 0° and 90° directions, the material tends not to be drawn into the die cavity. In contrast forbrass and copper type texture structures, the material isbest drawn into the die cavity at angles near 0° and 90°.
This shows that there are major differences in deformation depending on the texture structure of the material.
4.2

Comparison of Deformation of Deep Drawn
Products with Results of Material Tensile Tests
To investigate the relationship between the analytical results described above and the mechanical properties ofthe materials, 5052-O and 5182-O aluminum alloys were
subjected to uni-axial tensile tests and the relationshipbetween stress and strain was found. The tests were con-ducted using JIS #5 tensile test pieces at a tensile ratevirtually identical to the strain rate during ordinary deep
drawing.
The materials for the test pieces used in the uni-axial
tensile tests were 5052-300, 5052-12 and 5182-12 (where
the figures following the hyphens indicate average grain
size in µm). The decision to use these materials was basedon the results of measurements of the pole figures thatare described below.
Figure 8 shows {111} pole figures measured for the three
types of aluminum alloys. Observation of the pole figures
shows that 5052-12 is a brass-type alloy and 5182-12 isclose to copper-type. As for 5052-300, grain size was solarge (300 µm) that there was no orientation distribution
sufficient to discern the preferred orientation. Based, however, on a report8) that a cube-type texture structure appeared when annealed (recrystallized) after medium-reduction cold rolling, and on the fact that the effect of heat
treatment is greatly magnified when grain size is large, it
was concluded that the 5052-300 structure was close to
cube-type.
Thus although the materials differed, it is consideredappropriate from the standpoint of texture structure, to treat
the three alloys 5052-300, 5052-12 and 5182-12 as, respectively, a cube-type, brass-type and copper-type structure.
Figure 9 shows the relationship between true stress and

45
27
26

35

25

30

24

Cup height / mm

Flange radius / mm

40

25
20
15

23
22
21
20

10

19

5

18
17

0
0

45
Angle / deg

0

90

90

Figure 7 Relationship between wall height and angle fromrolling direction (B=1.6, punch stroke 28 mm)

Figure 6 Relationship between flange radius and angle from
β=2.0,punch stroke 27 mm)
rolling direction (βa) 5052-300

45
Angle / degb) 5052-12c) 5182-12

Figure 8 {111} pole figure

123Figure 9 Relationship between true stress and true straintrue strain found by uni-axial tensile tests for three typesof aluminum alloy sheets, in each of which the tension
direction was parallel to, at right angles to and at a 45°
angle to the rolling direction. Looking at the stress-strain
relationship, fracture strain was smallest in 5052-300 (close
to cube-type) test pieces of 45° direction, and strength
was high. In the 5052-12 (close to brass-type) and 518212 (close to copper-type) test pieces, on the other hand,fracture strain was high in those of 45° direction, andstrength was higher in those of direction parallel to and atright angles to the rolling direction.
From these tensile tests results, the following inferences
may be made regarding the deformation behavior of products made by cup deep drawing. In the 5032-300 (cubetype) test pieces, strength is higher for the 45° direction
and work piece elongation will not take place in that direction, and although the work is drawn into the die cavity, it
will be readily elongated in the rolling direction (0°) and in
the direction at right angles thereto (90°). It is therefore
inferred that earing will occur in the 0° and 90° directions,that is to say 0/90° earing will form. Conversely in the 505212 (brass-type) and 5182-12 (copper-type), strength in the
0° and 90° directions is higher than in the 45° direction, sothat in those directions the work will be drawn into the diecavity, elongation will be greater in the 45° direction, and
45° earing will form.
The fact that the hypothetical values obtained from tensile tests on the material agree with the analytical results
described above suggests that the analytical results obtained in this work are suitable.

and copper--analysis was conducted of the cup deep
drawing process, revealing a tendency toward 0/90°earing in the cube-type and 45°earing in the brassand copper-types.
(2) The results of the analysis agreed with those estimatedfrom tensile tests of the materials, suggesting the suitability of the analytical results.
It is planned to apply this analytical method in developing high-formability aluminum alloys.

REFERENCES

1) H.Morimoto and E.Nakamachi; Proc. 48th Joint Conf. of the Japan Soc. for Technology of Plasticity, (1997), 79. (in Japanese)
2) H.Morimoto and E.Nakamachi; Proc.1998 Spring Joint Conf. of
the Japan Soc. for Technology of Plasticity, (1998), 241. (in Japanese)
3) R.Hill; Mathematical Theory of Plasticity, Oxford Press, (1950),332.
4) J.W.Hutchinson; Proc. R. Soc. London, Ser.A, 384(1975), 101.
5) J.Pan and J.R.Rice; Int. J. Slolids Struct. 19-11(1983), 973.
6) Y.Zhou, K.W.Neal and L.S.Toth; Int. J. Plasticity, 9(1993), 961.
7) E.Nakamchi and X.Dong; Engineering Computations 13-2/3/
4(1996),327.
8) K.Itoh; J. of Japan Inst. of Light Metals, 43-5(1993), 285. (inJapanese)
Manuscript received on December 2, 1998.

5. CONCLUSION
The authors have developed a method of elastic/crystalline viscoplastic finite element analysis which adopts crystalline plasticity theory to define the evolution of anisotropy based on the elementary process of crystal slip, instead of conventional plastic potential theory, and applied
it to the deep drawing of aluminum alloys.
The influence of the texture structure of the material on
deformation of deep drawn products was investigated andthe following results were obtained:(1) Assuming three types of texture structure--cube, brass

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Furukawa Review, No. 18. 1999