Elastic-Plastic Flange Wrinkling of Circular Plates
in Deep Drawing Process
Farzad Moayyedian, Mehran Kadkhodayan
Department of Mechanical Engineering, Ferdowsi University of Mashhad,
Mashhad, Iran
fa_mo577@stu-mail.um.ac.ir, kadkhoda@um.ac.ir
Keywords: Elastic/plastic wrinkling, energy method, bifurcation functional, circular plate, deep
drawing process.
Abstract. This paper deals with two-dimensional plane stress wrinkling model of elastic/plastic
annular plate. Based on energy method and nonlinearity of strain-displacement law, a bifurcation
functional in polar coordinate is derived analytically. This technique leads to the critical conditions
for the onset of the elastic/plastic wrinkling of the flange during the deep-drawing process. Tresca
yield criterion along with deformation theory of plasticity are utilized and the material of the plate is
assumed to behave perfectly plastic. Moreover, the influence of the blankholder upon wrinkling and
on the number of the generated waves is quantitatively predicted by the suggested scheme. The
main advantage of the proposed solution is its better agreement with the experimental and analytical
results found by the other resarchers.
Introduction
Wrinkling is one of the major modes of the failure in automotive sheet pressing operations. This
behavior is caused by excessive compressive stresses during the forming. In a deep-drawing
operation an initially flat round blank is drawn over a die by a cylindrical punch, Fig. (1). The
annular parts of the blanks are subjected to a radial tensile stress, while in the circumferential
direction compressive stress is generated during drawing, Fig. (2). For particular drawing-tool
dimensions and blank thicknesses, there is a critical blank diameter/thickness ratio. The critical
stress causes the plastic buckling of the annular part of the blanks so that an undesirable mode of
deformation ensues with the generated waves in the flange [1, 2], Fig. (3). A bifurcation functional
was proposed by Hutchinson based on Hill general theory of uniqueness and also bifurcation in
elastic-plastic solids. This functional is given as;
1
F   ( M ij ij  N ij ij0  t  ij w,i w, j ) ds ,
(1)
2
where S denotes the region of the shell middle surface over which the wrinkles appear, w the
buckling displacement, t the thickness of the plate, N ij the force resultants, M ij the couple
resultants (per unit width),
 ij
the curvature tensor and  ij0 the stretch strain
tensor. This
bifurcation functional contains the total energy for wrinkling occurrence. On the other words, for
some non-zero displacement fields, the state of F  0 corresponds to the critical conditions for
wrinkles to occur [3]. In this paper, the complete form of this bifurcation functional is found by
using energy method. In plastic analysis, to have a closed-form analytical solution, Tresca yield
criterion and also deformation theory of plasticity are used. Therefore the complete form of the
functional (1) is suggested for elastic-plastic materials. The results show that the proposed solutions
for the critical conditions of the onset of the wrinkling have better agreement with the other avilable
experemental findings.
Elastic-plastic flange wrinkling
0
The nonlinear Lagrangian strain tensor is used in this paper is  ij   ij  z ij , where;
 0  u 0 1 w 2
 rr   r  2 ( r )

 0 u 0 1  v 0 1 1 w 2

 (
)
  
r
r 
2 r 

 0 1 1  u 0  v0 v0
w 1 w
 r  2 [ r     r  r  ( r ) ( r  ) ]


2w



,
 11
r2

1  w 1 2w


,
 22  
r  r r 2  2


1 2w
1 w
 2
.
 12  
r  r  r 

(2)
u 0 and v0 are the displacements in the middle plane of the plate, in r and  directions,
respectively. The buckling displacement normal to the plane of the plate in the z direction is shown
by w . The strain energy can be condensed in the following form for plane stress problem [4];
h
b 2 2
h
b 2 2
0
0
1
U  
2a

1
h ij ij dz r dr d  2 a

2
  (

rr
 rr       2 r  r ) dz r dr d .
(3)
h
2
The force ( N ij ) and the moment ( M ij ) resultants are found in terms of the displacement fields as;
h
h



2
2

0
0
N


dz

ij
ij
h
h Cijkl ( ij  z  kl ) dz  h Cijkl  ij ,




2
2

h
h



2
2
h3
0
M   z dz 
C
(


z

)
z
dz

C ijkl  kl .
ij
ij
ijkl
ij
kl
h
h

12



2
2
(4)
which C ijkl can be elastic or elastic-plastic stiffness matrix and h is the thickness of the plate. The
external force ( N̂ ij ) is acting in the middle plane of the plate (i.e. z  0 ). Considering this force,
and also using the first term of Eq. (2), it is easy to show that the external work can be written in the
following form [2-4];
b 2
b 2
b 2
(5)
0
0
WE    Nˆ ij ij rdrd     Nˆ ij ij0 rdrd     [ Nˆ 1111
 Nˆ 22  022  2 Nˆ 1212
]rdrd  .
a 0
a 0
a 0
Having the strain energy and the external work in hand and considering Nˆ ij  h ij , the total
potential energy function can be formed;
  U  WE  
b 2
1
h3
2w
1 w 1  2 w 2
{C1111( 2 ) 2  C 2222[ (

)]


2 a 0 12
r r r  2
r
 2 w 1 w 1  2 w
1  2 w 1 w 2
)[ (

)]  4C1212[ (

)] rdrd  }
2
2
r r r 
r r r 
r
b 2
u
u
1
1 w
1 v0 1 1 w 2 2
   h{C1111[ 0  ( ) 2 ]2  C 2222[ 0 
 (
) ] 
2a 0
r
2 r
r r  2 r 
 2C1122 (
u 0 1 w 2 u 0 1 v0 1 1 w 2
 ( ) ][ 
 (
) ]
r
2 r
r r  2 r 
v
v
1 1 u0
w 1 w 2
4C1212 ( [
 0  0  ( )(
)]) }rdrd 
2 r 
r
r
r r 
b 2
u
u
1 w
1  v0 1 1 w 2
   [h{ r [ 0  ( ) 2 ]    [ 0 
 (
) ]}rdrd 

r
2

r
r
r 
2 r 
a 0
2C1122[
(6)
e
Now the critical condition for elastic wrinkling can be obtained by substituting Cijkl
instead of C ijkl
for plane stress problem, in Eq. (6). To obtain the critical load and critical wave number the prebuckled elastic isotropic stress distribution in an annular plate subjected to a radial stress p along its
inner edge can be investigated [1] as;

p a2 b2


(  1) ,
 r
b2  a2 r 2


2
2
   p a (1  b ) .
 
b2  a2
r2

(7)
In this study, it is assumed that the displacement fields of the flange for a deep drawn cup have the
following form [10, 11];
 w(r, )  c (r  a) (1  cos n) ,

(8)
u 0 (r, )  d r cos n ,
 v (r, )  e r sin n ,
 0
where c , d and e are the constants and n is the number of generated waves in the flange [3]. It is
obvious that any admissible bifurcation mode in Eq. (8), satisfies the geometrical boundary
conditions u, v  0 and w  0 at the inner edge ( r  a ) and also the kinematical constraints
contain w(r , )  0 , u0 (r , )  0 and v0 (r , )  0 for a  r  b . By substituting the elastic stress
distribution from Eq. (7) and the displacements u 0 , v0 and w from Eq. (8) into the functional (6),
the following relationship may be found;

 c2D
2
E (m, n, ) 
1 b 2 hE
[ F (m, n, ) c 4  G (m, n, ) c 2 e 
2
64 (1   )
H (m, n, )c d  I (m, n, )d  J (m, n)e  K (m, n, )de] 
2
2
2
 c 2 b 2 tp
(9)
L(m, n).
2
The suggested functional may be shown in the following matrix form;

F  c c2
 M 11
 0
e 
 0

 0

d
0
M 22
M 32
M 42
0
M 23
M 33
M 43
0  c 
 
M 24  c 2 
 .
M 34   d 

M 44   e 
(10)
Now, the critical conditions for the onset of elastic wrinkling can be written as [2, 3];
 Det ( M ij )  0,


 [ Det ( M ij )]  0.
 n
(11)
The validity of the deformation theory in plasticity is limited to the monotonically increasing
loading in which:(1) the stress components are increased nearly proportionally in a loading process,
known as proportional loading; and (2) no unloading occurs. Hence, because these conditions are
acceptable in this case, it is preferred to use the deformation theory rather than the incremental
ep
theory of plasticity. For the perfectly plastic materials the Cijkl
is determined by the following
relationship [5];
f f
C epqkl
 mn   pq
,
f e f
Crstu
 rs
 tu
e
Cijmn
ep
e
Cijkl
 Cijkl

(12)
e
where f is a proper yield criterion suited for this problem and Cijkl
is the elastic coefficient of the
ep
hook’s law. To obtain Cijkl
, the Tresca yield criterion can be used as f  r    Y  0. In this
function, Y is the yield stress. For the plane stress problem (i.e.  33   13   23  0 ) a simple plane
stress relation is found. It should be noted that the plastic stress distribution in the flange before the
wrinkling is similar to the axisymmetric problem as [1];
b

 r  Y ln( r )  0,

  Y [ln( b )  1]  0.
 
r
(13)
ep
Utilizing this plastic stress distribution, displacement fields of Eq. (8) and obtained Cijkl
, we have;
t 3 c 2 E  ep
1 tE b 2
E
(
m
,
n
,

)

[ F ep (m, n, ) c 4  G ep (m, n) d 2  H ep (m, n)e 2
128 (1   2 )
96(1   2 )
F
(14)
t  b 2 c 2 Y ep
 I (m, n)de  J (m, n)c e  K (m, n)c d ] 
L (m, n).
8
ep
ep
2
2
The above functional can be written in matrix form similar to Eq. (10), and therefore the critical
conditions for the onset of wrinkling can be as Eq. (11). According to Fig. (1), when a spring-type
blankholder is used, assuming the spring coefficient of the blankholder as ( K ), its strain energy can
be written as
1
2
K (u 02max  v02max  wmax
) . To obtain the effect of blankholder upon onset of wrinkling
2
this energy should be added to Eq. (14).
Results and discussions
In this part, the critical conditions for the onset of elastic and plastic wrinkling in a deep drawn cup
are investigated for steel ( E  200 Gpa ,  0.3 ) and Aluminum ( E  70 Gpa ,   0.3 ). The second
condition of Eq. (11) yields to M 11  0 . From this equation ncr can be found and its variation with
n
a
) is shown in Fig. (4). After finding ncr , pcr is also calculated. A comparison between the
b
current obtained results and Yu’s results [1] has been shown in Fig. (5). It may be observed from
Fig. (6) that the Aluminum wrinkles under the lower loads than the steel for the same value of
a
( 1  ). The similar procedure can be used for plastic wrinkling. Figures (7) and (8) illustrate the
b
(1 
variation of n and
E t with ( a ) and the comparison of current results with some available
1
b
Y b
experimental data [1]. It can be observed that there is a good agreement between the suggested nonlinear solution and the experimental one. Figures (7) and (8) demonstrate the onset of wrinkling for
the flange and its comparison with the Geckeler and Senior results. It can be realized from Fig. (8)
that the Geckeler's result can give a good approximation for the small values of ( 1  a ), (i.e. narrow
b
flanges), but its error rises as the width of the flange increases [1]. It may be observed that for the
plastic wrinkling also Aluminum wrinkles under the lower loads than steel for the same value of
( 1  a ), Fig. (9). Moreover, increasing BHF (blankholder force) increases the number of waves for
b
the same value of ( 1  a ) and also postpone the onset of wrinkling, Figs. (10) and (11).
b
Conclusions
Using energy method, the elastic and plastic wrinkling of the flange in deep drawing process have
been studied analytically. The proposed bifurcation functional is more general than the ones of
previously proposed methods. A closed-form analytical solution is formed based on Tresca yield
criterion and deformation theory of plasticity and perfectly plastic material. The suggested technique
leads to the critical loads and the numbers of the generated waves. The influence of the blankholder
upon wrinkling and also on the number of the generated waves was quantitatively predicted by the
suggested scheme. The obtained analytical results have a good agreement with the experimental
data.
References
[1] T. X. Yu and W. Johnson: International Journal of Mechanical Sciences, Vol. 24 (1982), p.
175-88.
[2] E. Chu and Y. Xu: Journal of Mechanics and Physics of Solids, Vol. 43 (2001), p. 1421-1440.
[3] J. P. Corria and G., Ferron: Journal of Materials and Processing Technology, Vol. 155-156
(2004), p. 1604-1610.
[4] J. N. Reddy: Energy Principles and Variation Methods in Applied Mechanics (John Wiley &
Sons, 2002).
[5] A. Khan and S. Hung: Continuum theory of plasticity (John Wiley & sons, 1995).
Figure 1. Deep drawing process Figure 2. The flange is modeled
with cylindrical punch.
as an annular plate with radial
stress distribution in their inner
edges.
Figure 3. The waves produce in
the flange.
12
10
n
8
6
4
2
0
0
0.1
0.2
0.3
1-a/b
0.4
0.5
0.6
Figure. 4. The number of waves for the elastic wrinkling
of the flang.
Figure 5. The critical load for the elastic
wrinkling of the flange.
18
16
14
12
n
10
8
6
Present Result
Yu's Result
Experimental Results
4
2
0
Figure 6. Comparing the onset of elastic wrinkling for
steel and Aluminium.
Figure 8. The limitation of plastic wrinkling for the
flange.
Figure 10. Number of the generated waves in the flange with
Blankholder.
0
0.05
0.1
0.15
0.2
1-a/b
0.25
0.3
0.35
0.4
Figure 7. The number of generated waves in the
plastic flange wrinkling of the annular plate.
Figure 9. Comparing the onset of plastic wrinkling for
steel and Aluminium.
Figure 11. The onset of the plastic wrinkling
for different blankholder force(BHF).