Int J Adv Manuf Technol (2013) 67:877–885

DOI 10.1007/s00170-012-4532-2

ORIGINAL ARTICLE

Theoretical and experimental study on the effects of explosive

forming parameters on plastic wrinkling of annular plates
E. Kowsarinia & Y. Alizadeh & H. Samareh Salavati Pour

Received: 4 November 2011 / Accepted: 27 September 2012 / Published online: 10 October 2012
# Springer-Verlag London 2012

Abstract Wrinkling is increasingly becoming one of the while the other 20 % of the part failure is due to excessive
most common and troublesome modes of unacceptable de- tearing of various areas of the stamping [1]. The prediction
formation in sheet metal-forming prediction that is very and prevention of wrinkling are therefore extremely signifi-
important on the design of die geometry and processing cant in sheet metal operations. In spite of its practical impor-
parameters. In an effort to provide a reliable and efficient tance in sheet metal-forming applications, the wrinkling
tool to predict the critical blank holding force for prevention instability phenomenon has received very little attention in
of wrinkling, an analytical model for flange wrinkling in the industry. Much of the activity in the last two decades has
high-velocity forming processes, such as explosive forming, been directed toward analyzing tensile instabilities in sheet
is presented here. With consideration of constant blank metal-forming processes, and less attention has been given to
holder force and using a combination of energy method the compressive type of wrinkling instabilities.
and plastic bending theory, the critical radial displacement Naturally, wrinkling is a phenomenon of compressive in-
and number of wrinkling waves are obtained. For validation, stability. Geckler [2] provided a first approximate study of this
some experimental tests have been performed that their problem by considering the plastic regime, and his work was
results have adequate agreements with the analytical ones. then followed by many others. A considerable improvement
Moreover, the effects of process parameters such as blank of his results was achieved by Senior [3] who made several
holding force, radii ratio, and material mechanical properties assumptions regarding the deflected form of the plate and then
on wrinkling behavior has been discussed. evaluated the component quantities of the potential energy
related to a small deformation. By equating the total energy
Keywords Flange wrinkling . Explosive forming . Plastic tending to restore equilibrium to that due to forces displacing
bending . Blank holding force it, he found the critical stability conditions; this line of inquiry
resulted in a set of lower and upper bounds for both the critical
stress and the number of circumferential wrinkles.
1 Introduction Plastic bifurcation analysis is one of the most widely used
approaches to predict the onset of wrinkling. Hill’s bifurcation
Wrinkling is an undesirable phenomenon in sheet metal- and uniqueness theory [4] initiated the general analytical study
forming, especially when it occurs on flange where final part of plastic wrinkling. Furthermore, Hutchinson [5] developed a
appearance is important. Also, large wrinkles may damage bifurcation theory for structures in the plastic range where the
tools and interfere with part assembling and performance. Donnell–Mushtari–Vlasov plate and shell theory was used.
Eighty percent of the part failure in automotive pressings Hutchinson and Neale [6] later extended this work to study the
can be attributed to wrinkling of the binder (flange) or corner buckling behavior of doubly curved sheet metal; however, the
regions and/or the unsupported draw wall area of the part, investigation was limited to regions of the sheet that are free of
any contact, so the effect of the binder was not considered.
E. Kowsarinia :· Y.
E. Kowsarinia Y. Alizadeh (*):· H.
Alizadeh (*) H. S. S. PourSalavati Pour
Samareh Triantafyllidis and Needleman [7] studied this problem by
Department of Mechanical Engineering,
treating the binder as an elastic foundation and analyzed the
Amirkabir University of Technology,
P.O. Box 15875–4413, Tehran, Iran effect of binder stiffness on the critical buckling stress and
e-mail: alizadeh@aut.ac.ir wave number. Although their results compared favorably with
878 Int J Adv Manuf Technol (2013) 67:877–885

some previous empirical models for the cases without binder studied in previous works. Using energy method, the critical
constraint, no comparison with experimental results was given radial displacement and number of wrinkling waves are
for the cases with normal constraint. Another possible ap- obtained as functions of geometry, process kinematics, and
proach is the energy method. Senior derived flange wrinkling material mechanical properties. To validate the presented an-
criterion from energy considerations and a one-dimensional alytical technique, comparisons between this method and the
model that considered the effect of a spring-loaded blank results of experimental tests will be performed.
holder at a constant force [3]. Later, Yu and Johnson [8]
presented a two-dimensional buckling model of an elastic–
plastic annular plate resting on an elastic foundation to 2 Wrinkling model
determine the critical conditions and also quantitatively
investigated the effects of a binder on the critical buck- A thin annular plate of inner radius a, outer radius b, and
ling stress and wave number. Nevertheless, in the above thickness t ðt =b
1Þ is considered, corresponding to the
studies where the binder force is applied by springs or situation shown in Fig. 1. As usually, a cylindrical system of
by the stiffness of the holding down bolts of the blank coordinates (r, θ, z) is used to define various quantities of
holding plate, the binder force is a linear function of the interest associated to this problem. Displacement compo-
lateral flange surface deflection, and no direct correla- nents in the r, θ, and z directions, respectively, are denoted
tion between the binder stiffness and binder pressure, by ur, uθ, and w. The internal strains of interest are defined
which is used in the forming process, was provided. according to
Other than the above analytical approaches, numerical

@ur 1 @uθ
simulation using the finite element method (FEM) has be- "rr ¼ ; "θθ ¼ þ ur ;
come a useful tool to predict buckling behavior in the sheet @r r @θ

 ð1Þ
metal-forming involving complicated geometry and bound- 1 1 @ur @uθ uθ
"rθ ¼ þ  ;
ary conditions including friction. Cao and Boyce [9] pro- 2 r @θ @r r
posed a different approach to predict flange wrinkling under
normal restraints, where energy conservation and finite ele-
ment analysis were combined to obtain the critical buckling The linear pre-buckling state of stress in the annular plate
stress and resulting wavelength as functions of binder pres- is determined by solving the system of equilibrium equa-
sure. However, the use of FEM is somewhat time- tions of plane stress [12]
consuming, which is not convenient as a design tool. This
paper, instead of using the finite element analysis, devel- @σrr 1 @σrθ 1
ops an analytical approach to calculate the critical wrin- þ þ ðσrr  σθθ Þ ¼ g [ r ð2aÞ
@r r @θ r
kling conditions under a constant binder force and
@σrθ 1 @σθθ 2
pressure based on the same wrinkling criterion as Cao þ þ σrθ ¼ 0 ð2bÞ
and Boyce proposed. @r r @θ r
In addition, wrinkling or buckling instabilities are reduced by
high-velocity forming. This can be demonstrated in experiments
which were carried out by Padmanabhan [10]. In these tests, a
simple single-turn coil was used to compress 2-in.-diameter
aluminum and copper rings of various heights onto a 1-in.-
diameter-centered mandrel. This shows that, as the discharge 0-0-
b
energy increases, the rings become much more circular. In an-
other series of tests, aluminum and copper disks were accelerated
using a flat spiral coil at a truncated conical hardened steel form -
at various energies, without applying blank holding force. At low * +
energy, numerous wrinkles form in the sheets where line length a
must be reduced to conform to the shape. As energy increases, rr

they fit the form of the conical sections practically perfectly [11].
The present work is motivated by practical needs to anal-
ysis of flange wrinkling in high-velocity forming operations,
such as explosive forming, electromagnetic forming, etc., by t
applying the effect of blank holding force. Moreover, the
analysis of wrinkling in an annular plate with consideration Fig. 1 The flange is modeled as an annular plate subject to axisym-
of applying a constant blank holding force has not been metric radial stress
Int J Adv Manuf Technol (2013) 67:877–885 879

0.1 0.1

0.09 0.09

0.08 0.08

0.07 Wrinkling 0.07 Wrinkling

0.06 0.06
=0.1
0.05
0.05 =3.0
0.055 0.04
0.04
7.0
0.03
0.03 5.0
0.01 0.02
0.02
0.01 No Wrinkling
0.01
No Wrinkling 0
0 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Fig. 4 Normalized
 critical radial displacement versus radii ratio for
Fig. 2 Normalized
  critical radial displacement versus radii ratio for various K Sy
various q b2 Sy

Where γ is the density of plate material and ür is the "r ¼ "; "θ ¼ " ð5Þ
acceleration of plate which is encountered in high-velocity
Where " is the effective strain. To find the radial dis-
forming processes. Eqs. 2a and 2b must be supplemented
placement in the flat flange, the following displacement
with an appropriate set of boundary conditions. For simpli-
boundary condition is used.
fication, subscripts of normal strains and stresses will be
denoted by single notation. In this paper, it is assumed that
the pre-buckling state is axisymmetric, i.e., ur 0U(r) and ur jr¼ro ¼ uo ð6Þ
uθ00, for some function U [13],

@ It follows from Eqs. 4, 5, and 6 that the radial displacement

"rθ ¼ 0; σrθ ¼ 0; ðÞ ¼ 0 ð3Þ is inversely proportional to the current radius, i.e., ur ¼ ro uo =r.
@θ
Therefore, the effective strain is

Strain equations are written as follows: r o j uo j

"¼ ð7Þ
r2
@ur ur
"r ¼ ; "θ ¼ ; ð4Þ
@r r As σz 00, equilibrium equations become
For the proportional loading path on the facet of Tresca
yield surface in the range of σr >0 and σθ <0, the plastic radial @σr 1
þ ð σr  σθ Þ ¼ g [ r ð8Þ
and circumferential strains are given by the flow rules as @r r

20
19
18
17
16
15 =3.0
14 =0.1
13

12 0.055
11

10 7.0
9

8 0.01 7

6 5
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.4 0.6 0.8

  
Fig. 3 Wave numbers versus radii ratio for various q b2 Sy Fig. 5 Wave numbers versus radii ratio for various K Sy
880 Int J Adv Manuf Technol (2013) 67:877–885

0.1 0.1
0.09 0.09
0.08 0.08
0.07 Wrinkling 0.07 Wrinkling
0.06 0.4 0.06
0.05 0.05
0.05
0.04 0.04
0.03
0.03 0.25 0.03 =0.01
0.02 0.02
=0.1
0.01 0.01
No Wrinkling
No Wrinkling
0 0
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Fig. 6 Normalized critical radial displacement versus radii ratio for Fig. 8 Normalized critical radial displacement versus radii ratio for
various n various t =b

The boundary conditions are taken to be

σ ¼ σr  σθ ¼ Ko þ K "n ð10Þ
σr ðrÞ ¼ σ for *
r ¼ a; and
ð9Þ Where Ko and K are characteristic values of material and n
μq
σr ðrÞ ¼ σb ¼ for r ¼ b is the strain hardening exponent. In high-velocity forming
p bt processes, the value of K is dependent on strain rate, and its
Where μ is the friction coefficient and q is the blank influence will be discussed later. By considering of accelera-
holding force. The following Hollomon-type hardening re- tion ür to a constant m, using the boundary values, and after
lation between the effective stress and strain is considered substituting Eq. 10 in Eq. 8, solving of Eq. 8 gives

    
r 2n
n
σr ¼ K0 ln br þ K2nj ubonj μq
 1 þ m g ðr  bÞ þ pbt ;
 K j uo jn  r 2n 
b
 b ð11Þ
σθ ¼ K0 ln r  1 þ 2n bn b ð1  2nÞ  1 þ m g ðr  bÞ þ pμbtq

The deflection amplitude of the flange is taken in the wrinkles, respectively. The occurrence of wrinkling is inevita-
form [14] bly accompanied by a radial reduction of the plate for geomet-
w ¼ wA ðr  aÞð1 þ cos N θÞ; ð a  r  bÞ ð12Þ rical reasons. Note that the length of the wrinkled rim is equal to
the initial circumference of the blank since wrinkling is as-
Where r, N, and wA are the radial position, the number of sumed to begin from the onset of loading. Therefore, the length
waves, and a constant associated with the amplitude of of the wrinkled rim should satisfy the following equation

20 30

18
25
16
0.25
14 =0.1 20 =0.02
12

10 15
0.03
8
10
6 0.05
0.4
4
5
2

0 0
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Fig. 7 Wave numbers versus radii ratio for various n Fig. 9 Wave numbers versus radii ratio for various t =b
Int J Adv Manuf Technol (2013) 67:877–885 881

Table 1 The mechanical properties of AA 5456 aluminum

Ultimate stress (MPa) Yield stress (MPa) Percentage of elongation Strain hardening exponent Strength constant, K (MPa)

348.2 148.6 25.9 0.224 494.7

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2ffi
Z Z 2p
@w N wA ðr*  aÞ sinðN θÞ
l¼ 1þ ds ¼ 1þ r* dθ ¼ 2p b ð13Þ
@s 0 r*

R1  
Where ds is the infinitesimal arc length of the wrinkled rim T1 ¼ ρ σr dρ ¼ K0 14 ð1  a2 Þ þ a2 ln a
2
a
projected on the flat initial blank plane and r* is the current
 22n   3 
blank radius defined as ðb  ju0 jÞ . After some simplifying n
þ K j2nuo j 1a 1a2 1a2
2ð1nÞ  2 3  2
þ g m 1a ð17Þ
treatment under the assumption of constant thickness, t0t0,  2
and using the Taylor expansion of Eq. 13, the deflection þ μp qt 1a 2
amplitude of wrinkles shape is obtained as follows:
p pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi And
wA ¼ juo jð2  3juo jÞ ð14Þ
N ð1  a Þ
R1 Þ2
The work done by the stress distribution in Eq. 11 is T2 ¼ a σθ ðρa dρ ¼
h ρ i
K0 14 ð1  a2 Þ þ a2 ln a  a2 ðln aÞ2
2 2
obtained from [15]
h i
R 2p R b h  2  1 @w 2 i
n
þ K j2nuo j 212n
ð1nÞ ð1  a
22n
Þ  a2 ð1þ2n Þ
2n ð1  a
2n
Þ  12 ð1  a2 Þ þ a2 ln a
h 3 i
W ¼  2t 0 a σr @w @r þ σ θ r @θ r dr dθ; ð15Þ þg m 1a ð1aÞ2
þ a2 ln a þ μq
3  pt 2 ð1  aÞð1  3aÞ  a ln a
1 2
2

Integration of Eq. 15 gives ð18Þ

t b2 w2A p   In the previous equation, the dimensionless parameters

W ¼ 3 T1 þ N 2 T2 ð16Þ
2 a ¼ a=b and ρ ¼ r=b are used. As shown in Yu and Johnson
[6], the strain energy due to bending deflection in the case of
Where proportional loading paths takes the form of

Fig. 10 The experimental

arrangement of free explosive
forming die
882 Int J Adv Manuf Technol (2013) 67:877–885

R 2p R b  @ 2 w @w @2 w
2    2
 @@rw2 1 @w @2 w 1 @2w @w
2
U¼ D
2 0 a @r2 þ 1r @r þ r12 @θ2 r @r þ r12 @θ2
þ r @r@θ  r12 @θ r dr dθ ð19Þ

where D is the bending rigidity defined as Eo t 3 =9 [16]. E0 is 4EEt

E0 ¼ pffiffiffiffi pffiffiffiffiffi2 ð20Þ
the reduced modulus of the rectangular cross-section which E þ Et
can be written as a function of the Young’s modulus, E, and
the tangent modulus, Et. The result of integral 19 is

p D w2A  1  
U¼ 2 3 ln a þ 2 ða  1Þð3  aÞ  ln a ð1 þ N 2 ÞN 2  4N 2 ða  1  ln aÞ  N 2 ln a ð21Þ

R R    
And the potential energy of total system is expressed as [17] WF ¼ q ðp ðb2  a2 ÞÞ dw ds ¼ 2wA q bð1  aÞ
ð23Þ
Π ¼ WF þ U  W ð22Þ
In order for the system to be in equilibrium and wrinkling
has not occurred, the minimum total potential energy must
Where WF is the external work of blank holding force always be positive [18]. By taking Π00, the value of radial
and is obtained as follows displacement in outer edge of plate, j uo j, is obtained as

Table 2 The parameters of tests and theoretical and experimental results

Plate outer Mass of explosive Standoff Blank holding Theoretical radial Theoretical Experimental Experimental
radius (mm) charge (gr) distance force (kN) displacement of number of displacement of number of
(mm) outer edge (mm) wrinkling waves outer edge (mm) wrinkling waves

90 10 300 19.28 1.45 29 1.6 23

90 10 150 31.43 2.85 30 2.6 25
90 20 400 42.93 4.5 31 4.1 27
90 20 300 46.66 5.1 32 4.7 29
100 10 300 19.78 0.9 26 0.8 24
100 10 150 34.20 1.9 27 1.7 23
100 20 400 49.20 3.2 29 3.1 22
100 20 300 54.20 3.7 30 4.1 26
110 10 300 21.37 0.7 25 0.7 22
110 10 150 34.61 1.35 26 1.5 23
110 20 400 48.40 2.15 28 2.4 22
110 20 300 53.72 2.5 28 2.3 25
120 10 300 18.78 0.45 23 0.5 19
120 10 150 36.58 1.1 25 1.0 21
120 20 400 46.08 1.5 26 1.2 19
120 20 300 51.26 1.75 26 1.5 19
130 10 300 18.25 0.35 22 0.4 16
130 10 150 32.45 0.75 23 0.9 20
130 20 400 43.26 1.1 24 0.9 20
130 20 300 47.66 1.25 25 1.0 21
140 10 300 15.99 0.25 21 0.2 16
140 10 150 29.52 0.55 22 0.4 18
140 20 400 39.20 0.8 22 1.0 19
140 20 300 42.73 0.9 23 0.7 21
Int J Adv Manuf Technol (2013) 67:877–885 883

Fig. 11 Some of experimental

test specimens

  1=n And
t w2A p    
j uo j ¼ WF þ U þ 3 T3 þ N T4 =T5
2
1  a2 a2
2 T4 ¼ K0 1  a2 þ ln a  ðln aÞ2
4 2 2
Where ð24Þ " #
2
1  a 3 ð1  a Þ
   3  þ g mb  þ a ln a
2
ð26Þ
T3 ¼ K0 1
ð1  a2 Þ þ a2 ln a þ g mb 1a
2
 1a2 3 2
4 3 2  
 2 ð25Þ μq 1
þ p b2μðqt=bÞ 1a þ 2 ð1  aÞð1  3aÞ  a ln a
2
2 p b ðt =bÞ 2

0.05
0.07 (Predicted)
(Predicted) 0.045
(Experimental)
0.06 (Experimental) 0.04 (Predicted)
(Predicted)
0.05 0.035 (Experimental)
(Experimental)
0.03
0.04
0.025

0.03 0.02

0.015
0.02
0.01
0.01
0.005

0 0
0 0.005 0.01 0.015 0.02 0.025 0.03 0 0.005 0.01 0.015 0.02 0.025 0.03

Fig. 12 Comparison of critical radial displacement in outer edge Fig. 13 Comparison of critical radial displacement in outer edge
between theoretical and experimental results between theoretical and experimental results
884 Int J Adv Manuf Technol (2013) 67:877–885

0.04
And (Predicted)
0.035
(Experimental)
n  22n 
t w2 p 1a2 0.03 (Predicted)
T5 ¼  2A  3K 2ð1nÞ  2
1a
h 2n
io (Experimental)
Þ  a2 ð1þ2nÞ 2n
2 0.025
ð1nÞ ð1  a
þ N2nK 212n 22n
2n ð1  a Þ  12 ð1  a2 Þ þ a2 ln a
0.02
ð27Þ
0.015

0.01
By solving nonlinear Eq. 24, radial displacement at the
outer edge of plate, u0, can be evaluated for any prescribed 0.005

wrinkling mode (N). The minimum of these values can prevent 0

wrinkles for all possible wave numbers, and it is defined as the 0 0.005 0.01 0.015 0.02 0.025 0.03

critical radial displacement in outer edge, ucr

0 [19]. Therefore,
 cr 
u  ¼ minfjuo ðN Þj for all N 2 ½1; 1 g ð28Þ Fig. 14 Comparison of critical radial displacement in outer edge
o
between theoretical and experimental results

3 Results and discussions 4 Verification and discussion

Figures 2 and 3 illustrate the dependence of the normalized

  For the purpose of experimental verification, AA5456-O
critical radial displacement in the outer edge, ucr 
o =b , and (1.0 mm thickness) was prepared. Properties of this sheet mate-
wave number, Ncr, on the normalized blank holding force, rial are necessary for solving Eq. 28. The hardening behaviors
   
q b2 Sy . The parameters n00.2, K0 Sy ¼ 0, K Sy ¼ 5:0, were approximated by the hardening law shown in Eq. 29,

t =b ¼ 0:025, and g m b Sy ¼ 0:3 are used in generating obtained from the uni-axial conventional tension tests. The me-
the curves, where Sy is the yield stress of plate material. It chanical properties which were determined in the tensile tests are
can be seen that the normalized blank holding force given in Table 1. As mentioned earlier, the value of K is depen-
  dent on strain rate, and its influence was discussed previously.
q b2 Sy strongly affects the wrinkling resistance. An in-
   
crease in q b2 Sy strongly increases ucr 
o =b and also σ ¼ 503:5 "0:21 ðMPaÞ ð29Þ
increases the resulting wave numbers.
Figures 4 and 5 correspond to the critical conditions In this study, a free explosive forming die with the hole
with various material strength coefficients K, where diameter of 140 mm was used, where its arrangement is
    illustrated in Fig. 10, where the diameter of the hole in the
q b2 Sy ¼ 0:05 . An increase in K decreases ucr 
o =b
die was 140 mm. The hexogen (RDX) was used as explosive
and also slightly decreases the resulting  wave  numbers.
Figures 6 and 7 show the dependence of ucr 
o =b and wave 35

numbers on the strain hardening exponent n. As n increases, the

30
critical radial displacement increases and wave numbers
decrease. 25

Figures 8 and 9 depict

 the effect of the normalized plate
thickness t =b on ucr =b and wave numbers. It is seen that t =b
20

o
15
has essentially little influence on the critical wrinkling con-
(Predicted)
ditions. But a decrease in t =b significantly increases the 10
(Experimental)
wave numbers. The same result has been obtained by
5 (Predicted)
Agrawal et. al. [20]. They reported that as the sheet thick- (Experimental)
ness increases the energy required to bend the sheet for the 0
0 0.005 0.01 0.015 0.02 0.025 0.03
given number of wrinkles increases, hence the number of
wrinkles decreases. Figures 2 to 9 show that the absolute
value of critical radial displacement and number of waves Fig. 15 Comparison of number of wrinkling waves between theoret-
increase significantly with increasing of radii ratio a=b. ical and experimental results
Int J Adv Manuf Technol (2013) 67:877–885 885

charge, which is shaped in cylindrical form by pressing. By model is its ability to obtain accurate results at an extremely low
using a 2D finite element code, the simulation was performed, computational cost. It will be attractive to the design commu-
and the acceleration applied on the flange was obtained and nity where quick turnaround is an essential requirement. The
used as the input parameter of the analysis. By using the experimental investigation on more different materials and
presented analysis, the theoretical radial displacements of process parameters can be suggested for further works.
outer edge, for a known blank holding force, were obtained.
By explosive forming tests, the experimental values of radial
displacement of flange edge and number of wrinkling waves
were obtained and compared with the predicted results (Ta- References
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