Finiteelement simulation of springback in sheet metal forming using local interpolation for tool surfaces
ABSTRACT This paper describes new techniques for the sheet metal forming simulation using a local interpolation for tool surfaces proposed by Nagata [Simple local interpolation of surfaces using normal vectors. Computer Aided Geometric Design 2005;22:327–47] and the effect of tool modeling accuracy on springback simulation of a high strength steel sheet. The Nagata patch enables the creation of tool models that are much more accurate, in terms of not only shape but also normal vectors, than those of conventional polyhedral representations. Besides allowing an improved description of the contact between the sheet nodes and the tool surfaces, the proposed techniques have the advantage of relatively straightforward numerical implementation. Springback simulations of a twodimensional draw bending process of a high strength steel sheet are then carried out using the polyhedral and Nagata patch models. It is found that the simulation results are largely influenced by the tool mesh when using polyhedral representations, while they are rather independent when using the Nagata patch representations. This demonstrates the efficiency and reliability of the numerical solution using the Nagata patch model.

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ABSTRACT: This paper describes the effect of tool discretization on the simulation of metal forming processes, especially for processes where the contact area is quite small with respect to the component size. The smoothing of contact surfaces, which are defined by linear triangles, is based on a higher order quadratic interpolation of the curved surface. This interpolation is derived from the node positions and their normal vectors, as proposed by Nagata. The normal vectors are calculated at each node from the existing discretized surface by considering a patch of surrounding elements and using a consistent strategy. The efficiency and reliability of the resulting contact model are assessed on several examples, such as the indentation of a parallelepiped and the drawing of a wire.07/2011: pages 2340;  [show abstract] [hide abstract]
ABSTRACT: This study deals with the new strategy currently implemented in DD3IMP inhouse code to describe the forming tools using Nagata patches. The strategy is based on the use of the Nagata patch interpolation to generate smooth contact surfaces over coarse faceted finite element meshes. The description of the adopted algorithm is briefly presented, highlighting the contact search algorithm employed. The reverse deep drawing of cylindrical cups, proposed as benchmark at the Numisheet’99 conference, is selected to examine the accuracy and robustness of the proposed approach. The effect of the gap between the blankholder and the die is studied, adopting two distinct strategies: fixed gap and variable gap. The numerical results are compared with the experimental ones, previously presented and discussed in [1]. It is shown that the agreement is very good both in terms of punch force evolution and thickness distribution.Key Engineering Materials 07/2013; 554557:22772284.
Page 1
International Journal of Mechanical Sciences 50 (2008) 175–192
Finiteelement simulation of springback in sheet metal forming
using local interpolation for tool surfaces
Takayuki Hamaa,b,?, Takashi Nagatab, Cristian Teodosiub,c,
Akitake Makinouchib, Hirohiko Takudaa
aGraduate School of Energy Science, Kyoto University, YoshidaHonmachi, Sakyoku, Kyoto 6068501, Japan
bVolumeCAD System Research Program, The Institute of Physical and Chemical Research, 21 Hirosawa, Wako, Saitama 3510198, Japan
cLPMTMCNRS, University Paris 13, 93430 Villetaneuse, France
Received 11 April 2007; received in revised form 6 July 2007; accepted 8 July 2007
Available online 13 July 2007
Abstract
This paper describes new techniques for the sheet metal forming simulation using a local interpolation for tool surfaces proposed by
Nagata [Simple local interpolation of surfaces using normal vectors. Computer Aided Geometric Design 2005;22:327–47] and the effect
of tool modeling accuracy on springback simulation of a high strength steel sheet. The Nagata patch enables the creation of tool models
that are much more accurate, in terms of not only shape but also normal vectors, than those of conventional polyhedral represen
tations. Besides allowing an improved description of the contact between the sheet nodes and the tool surfaces, the proposed techniques
have the advantage of relatively straightforward numerical implementation. Springback simulations of a twodimensional draw bending
process of a high strength steel sheet are then carried out using the polyhedral and Nagata patch models. It is found that the simu
lation results are largely influenced by the tool mesh when using polyhedral representations, while they are rather independent when
using the Nagata patch representations. This demonstrates the efficiency and reliability of the numerical solution using the Nagata
patch model.
r 2007 Elsevier Ltd. All rights reserved.
Keywords: Sheet metal forming; Springback; Finite element method; Contact; Tool modeling; Quadratic polynomial patch
1. Introduction
In the automotive and die manufacturing industries,
prior predictions of the formability of sheet metals using
finiteelement methods (FEM) have enabled lowering cost
and shortening delivery periods, as well as the achievement
of a higher quality [1–4]. Their contribution is recently
increasing and became indispensable in many industrial
applications. Prediction accuracy for shape defects includ
ing springback and surface deflection is, however, still
inadequate, and empirical trialanderror methods are
presently used in the field operations. Moreover, high
tensile strength steels are nowadays commonly used, which
makes the amount of springback remarkably large, and
renders the prediction of springback much more important.
Therefore, the establishment of simulation technologies for
improving the prediction accuracy is an urgent and
challenging task.
For the purpose of dealing with the above problems,
many attempts have been made to improve the accuracy of
sheet metal forming simulation by FEM over the last
decade, including the development of new finite elements
[5–13], the improvement of material constitutive models
[14–21], and the coupled analysis of the elastoplastic sheet
deformation with the elastic deformation of tools [22,23].
Although the treatment of the contact between a sheet and
tools is one of the most important issues in order to achieve
accurate simulations, attempts to improve the tool model
ing accuracy are scarce [24,25], thus the effect of tool
modeling accuracy on simulation is not clear.
ARTICLE IN PRESS
www.elsevier.com/locate/ijmecsci
00207403/$see front matter r 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijmecsci.2007.07.005
?Corresponding author. Graduate School of Energy Science, Kyoto
University, YoshidaHonmachi, Sakyoku, Kyoto 6068501, Japan.
Tel.: +810757535418; fax: +810757535428.
Email address: hama@energy.kyotou.ac.jp (T. Hama).
Page 2
For sheet metal forming simulations, the tools are in
general assumed rigid and only their surfaces are modeled
by either a set of polyhedral patches (i.e., finiteelement
meshes), points, or parametric representations [24–26].
Although the mesh approach is employed in most cases due
to its capability of wide application and simplicity in use,
this approximation sometimes leads to large errors in the
geometry [27,28]. Since the contact formulation generally
considers the tools approximated as polyhedral surfaces
[29–34], such geometrical errors may affect in turn the
simulation accuracy [35]. For instance, considering simula
tions of surface deflection, it is generally estimated that the
elastic recovery, as main origin of this defect, should be
predicted with the accuracy of the order of 10mm, since the
defect itself is the order of dozens to several hundred mm
[4,36–38]. Clearly, tool models with higher level of
accuracy should be used for this purpose.
In order to improve the contact surface, various
techniques to smooth the discretized surfaces, such as
NURBS, Be ´ zier patch, and Gregory patch, have been
proposed by many researchers [35,39,40]. The conventional
smoothing involves free parameters or assumptions on the
derivatives, which should be given a priori. This is,
however, hardly possible since the analytic properties of
the original surface are generally unknown [27]. Therefore,
the geometrical accuracy of the conventional smooth
surface interpolators is not assured, and has not even been
studied [27]. Furthermore, since the smoothed surfaces are
generally described by cubic or higher polynomials or
rational functions, the contact search requires significant
effort, which is sometimes the central factor for increasing
computational time and instability [30].
Recently, Nagata [27] has proposed a simple algorithm
for interpolating discretized surfaces to recover the original
geometry with good accuracy (subsequently called the
Nagata patch). The Nagata patch is a quadratic parametric
interpolator determined only by the position and normal
vectors given at the vertices of polyhedral meshes. The
algorithm involves no free parameters, and the interpolated
surface rapidly converges to the original one when
decreasing the mesh size. The Nagata patch yields an
accurate and simple geometrical description that can
hardly be obtained by the standard polyhedral approxima
tions without drastically increasing the number of tool
elements.
For the purpose of improving the tool modeling
accuracy and clarifying its effect on the quality of
numerical solution, this paper proposes new techniques
of the sheet metal forming simulation using the Nagata
patch for describing the tool surfaces. A contact search
algorithm and a consistent contact tangent stiffness matrix
considering tool models represented by the Nagata patch
are formulated. Simulations of a twodimensional draw
bending process are then carried out using both tool
models represented by the Nagata and polyhedral patches,
and the effect of the tool modeling accuracy on the
springback simulation is examined in detail.
2. Finiteelement formulation
2.1. Virtual work and material constitutive equations
New techniques of the sheet metal forming simulation
using the Nagata patch for the tool surfaces are introduced
in the static explicit FEM code STAMP3D [5,41]. The
formulation applied in STAMP3D may be briefly described
as follows. An updated Lagrangian rate formulation is
used to describe the finite deformation. The rate form of
the equilibrium equations and boundary conditions at time
t are equivalently expressed by the principle of virtual
velocity in the form:
Z
where V and S denote, respectively, the domain occupied
by the body and its boundary at time t. S1is the part of the
boundary S on which the nominal surface traction f is
prescribed, while S3is that on which contact conditions
between the sheet and the tools are prescribed. dv is the
virtual velocity field satisfying the conditions dv ¼ 0 on the
velocity boundary S2 and dvN
boundary S3, where the subscript N denotes the normal
component. P is the first Piola–Kirchhoff stress tensor.
The superscript T denotes the transpose. Assuming that all
the rateform relations are preserved from time t to t+Dt,
where Dt is a small time increment, the incremental form of
Eq. (1) can be derived as
Z
¼
S1
S3
Z
or
Z
¼ Dt
S1
S3
Z
where the following relations:
V
_PT: dv ? rx
ðÞdV ¼
Z
S1
_f ? dvdS þ
Z
S3
_f ? dvdS;
(1)
¼ 0 on the contact
V
D sJ
Z
þ
?? 2r ? DD
??: dD þ DL ? r
Df ? dvdS þ
Z
ðÞ : dL
??dV
f ? dvdS
Df ? dvdS þ
ZZ
S1
S3
f ? dvdS ?
V
r : dLdV,
ð2Þ
Dt
V
sJ
Z
?? 2r ? D
??: dD þ L ? r
_f ? dvdS þ Dt
ðÞ : dL
??dV
Z
r : dLdV,
_f ? dvdS þ
Z
S1
f ? dvdS
þ
S3
f ? dvdS ?
Z
V
ð2Þ0
Df ¼_f Dt
and
DP ¼_PDt ¼ DsJ
are introduced. r is the Cauchy stress tensor sJ
Jaumann rate of the Kirchhoff stress tensor, L is the
velocity gradient tensor, and D is the strain rate tensor,
which is the symmetric part of L. The third, fourth, and
fifth terms of the righthand side in Eq. (2) remain in order
to cancel the nonequilibrated forces arising from the
(3)
?? DD ? r ? r ? DD þ r ? DLT
(4)
?is the
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T. Hama et al. / International Journal of Mechanical Sciences 50 (2008) 175–192
176
Page 3
explicit timemarching algorithm, as explained in the next
section.
Smallstrain linear elasticity and large deformation, rate
independent workhardening plasticity are assumed for the
constitutive equations. In order to take into account the
anisotropy of sheet, Hill’s quadratic yield function [42] and
the associated flow rule with isotropic hardening are used.
The elastoplastic constitutive equations are given in the
form:
sJ
?¼ Cep: D ¼ Cep: L,
where Cepis the tensor of the tangent elastoplastic moduli.
(5)
2.2. Discretization
A fournode isoparametric shell element [5,43] is
employed in this study. Assembling the discretized virtual
work over the entire domain, the tangent stiffness equation
is obtained as
?
where [K] is the tangent elastoplastic stiffness matrix, {Du}
is the displacement increment vector, {Df} is the equivalent
nodal force increment vector due to the surface traction
and {Dfc} is that due to the contact, which is formulated in
the next chapter. {Fneq} is the nonequilibrated nodal force
vector, i.e., the difference between the external and internal
forces. The tangent stiffness matrix [K] is calculated at time
t and considered as constant over the increment Dt. To
preserve the approximate linearity during the increment,
the generalized rminstrategy [44] is employed to appro
priately limit the size of the increment.
Using the tangent solution described above, the nodal
equilibrium between the external and internal forces at the
end of increment is not guaranteed and the nonequili
brated force vector {Fneq} remains. To overcome the
difficulty, a special algorithm called ‘‘ALGONEQ’’ [45–47]
is employed in STAMP3D, in order to systematically
cancel the remaining nonequilibrated nodal forces when
ever their norm exceeds a prescribed admissible tolerance.
K
½ ? Du
fg ¼ Df
?þ Dfc
??þ Fneq
fg,(6)
3. Contact treatment
3.1. Virtual work
The masterslave algorithm is employed, in which sheet
nodes and tool elements are considered to be slave nodes
and master segments, respectively. The tools are considered
rigid and their surfaces are discretized by a set of triangular
patches. The virtual work due to the contact, which is the
second term in the righthand side of Eq. (2)0, can be
written as
Z
S3
_f ? dvdS
??
Dt ¼
Z
S3
_^fN^ e3þ^fN_^ e3þ_^f
i
T^ eiþ^f
i
T_^ ei
??
? dvdS
??
Dt,
(7)
where ^ ej, j ¼ 1, 2, 3, denote the base vectors of a local
Cartesian frame defined at each contact point, where the
unit vector normal to the tool surface at the contact point
is taken as ^ e3, while ^ e1and ^ e2lie within the tangent plane.
The superscripts4denotes the components with respect to
this local frame, the subscripts N and T denote the normal
and tangent components of the contact force, respectively,
and i ranges over 1 and 2. For simplicity, friction is not
taken into consideration in this study so that the last two
terms of the integrand in the righthand side of Eq. (7) are
omitted in the following. Therefore, the contact force f
should have only a normal component and thus Eq. (7)
yields
Z
Since a penalty method is employed to satisfy the
impenetrability condition in this study, the condition
d^ vN¼ 0 no longer holds and Eq. (8) is discretized in a
rather standard way (see e.g. Ref. [48]) as
S3
_f ? dvdS
??
Dt ¼
Z
S3
_^fN^ e3þ^fN_^ e3
??
? dvdS
??
Dt.(8)
Z
S3
_f ? dvdS
??
Dt ¼
X
X
l
l_f
c? dlvDt
¼
l
al^ vrel
Ndl^ vNþl^f
c
N
l_^ e3? dlv
??
Dt,
ð9Þ
where l denotes the number of the contact nodes,l_f
concentrated contact force vector at node l,
normal component of the relative velocity of sheet node l
with respect to the tool, and a is the penalty number. The
second term in the righthand side of Eq. (9) arises from
rotations of the contact force vectors when the contact
nodes are sliding on curved parts of the tool surface.
cis the
Nis the
l^ vrel
3.2. Contact treatment using the Nagata patch for tool
surfaces
The Nagata patch is employed for the tool surfaces
discretized by a set of triangular patches. In this section,
the formulation of the Nagata patch is briefly described in
Section 3.2.1, followed by Sections 3.2.2 and 3.2.3
formulating contact search algorithms between sheet nodes
and tool elements (the Nagata patch). The second term in
the righthand side of Eq. (9) is then discretized and the
consistent tangent stiffness matrix for the contact taking
into account the Nagata patch is derived in Section 3.2.4.
3.2.1. Nagata patch formulation
The Nagata patch is a quadratic parametric interpolator
for polyhedral meshes [27]. The algorithm recovers the
curvature of surfaces using the position and tool normal
vectors at the element nodes in the following procedure. It
is noted that the normal vectors that are consistent with
original CAD surfaces are supposed to be given at the
element nodes.
ARTICLE IN PRESS
T. Hama et al. / International Journal of Mechanical Sciences 50 (2008) 175–192
177
Page 4
(a) Interpolation of edges of a tool element
An edge of a tool element is replaced by a curve
given in the form:
x x ð Þ ¼ x1þ x2? x1? c
where x is a parameter satisfying the condition
ðÞx þ cx2,(10)
0pxp1.(11)
x1and x2are the position vectors of the ends of the
edge (i.e., the element nodes), and c is the vector
adding curvature to the edge. Assuming that the curve
given by Eq. (10) is orthogonal to the normal vectors
n1and n2given at nodes x1and x2, respectively, and
minimizing the norm of c, the vector c can be
determined as
c ¼
n1;n2
1 ? c2
n1;?n1
2
½?
1
?c
1
?c
?
??
n1? x2? x1
?n2? x2? x1
Þ
Þ
ðÞ
ð
)
Þ
()
ca ? 1
ðÞ
½
n1? x2? x1
?n1? x2? x1
ð
ð
(
¼ 0
c ¼ ?1
ðÞ
8
<
>
>
>
>
>
>
>
>
>
>
:
,
(12)
where c ¼ n1?n2. All the edges of the tool element are
replaced by the curves given by Eq. (10).
(b) Interpolation of the surface of a tool element
Curvatures of a tool element can be recovered by
interpolating the curves given by Eq. (10). In the case
of a triangular patch, the interpolated surface is given
by the quadratic polynomial:
x Z;z
ðÞ ¼ C1þ C2Z þ C3z þ C4Zz þ C5Z2þ C6z2,
(13)
where Z and B are the parameters defined on the surface
element region, and satisfy the next condition:
0pzpZp1.(14)
The coefficient vectors are given by
C1¼ x1;
C3¼ x3? x2þ c1? c3,
C4¼ c3? c1? c2;
where c1, c2, and c3 are the vectors defined by
Eq. (12) for the edges (x1, x2), (x2, x3), and (x3, x1),
respectively.
By replacing all the triangular patches by the Nagata
patch (Eq. (13)), the discretized tool surface can be
more precise without increasing the number of
elements. The reader is referred to the original paper
[27] for details concerning the Nagata patch and the
assessment of its accuracy. Techniques for detecting
C2¼ x2? x1? c1,
C5¼ c1;
C6¼ c2,
ð15Þ
contact conditions between the Nagata patches and the
sheet nodes, i.e., the contact search algorithm, and for
the consistent contact tangent stiffness matrix taking
into account the Nagata patch are newly formulated,
and are described in Sections 3.2.2, 3.2.3, and 3.2.4.
3.2.2. Contact search algorithm for free nodes
In STAMP3D, the contact search for free sheet nodes is
done by tentatively calculating the intersection point xcof a
relative displacement increment vector Durelof a free sheet
node x0with a tool element [32]. It should be noted that,
when using shell elements for the sheet, the assumed
surface halfthickness away from the midsurface of the
shell element x0
account the thickness of the sheet in the contact search [49],
where
his used instead of x0in order to take into
xh
0¼ x0?h
where V0is the fiber vector of the shell element at node x0
and h is the length of V0. The sign is determined by which
side of the surface is considered. The contact search for free
nodes is schematically shown in Fig. 1, where x0
denote the position vectors of the assumed sheet node xh
the beginning and end of the increments, respectively.
First, the local Cartesian frame f~ e1; ~ e2; ~ e3g is defined as
Durel
Durel
~ e3? C6
where ~ e1can arbitrarily be chosen in the plane perpendi
cular to ~ e3if ~ e3? C6¼ 0. The sheet node never intersects
the Nagata patch if Durel¼ 0, and hence is being
considered to remain free without conducting the following
calculation. Writing Eq. (13) in a component form with
respect to f~ e1; ~ e2; ~ e3g yields
C1
C2
C3
2V0,(16)
tand x0
t+Dt
0at
~ e3¼
????;
~ e1¼
~ e3? C6
kk;
~ e2¼ ~ e3? ~ e1,(17)
1? x1
1? x2
1? x3
where the superscripts of the coefficients denote their
components, and xc1, xc2, and xc3are the components of xc.
The condition C6
definition of ~ e1. Since the relative displacement increment
cþ C1
cþ C2
cþ C3
2Z þ C1
2Z þ C2
2Z þ C3
3z þ C1
3z þ C2
3z þ C3
4Zz þ C1
4Zz þ C2
4Zz þ C3
5Z2¼ 0,
5Z2þ C2
5Z2þ C3
6z2¼ 0,
6z2¼ 0,
ð18Þ
1¼ 0 is always attained due to the
ARTICLE IN PRESS
e1
~
e2
~
e3
~
x0t
x1
x2
x3
Nagata patch
Triangular patch
xc
Δurel
x0t+Δt
Fig. 1. Schematic of the contact search between a free node and the
Nagata patch.
T. Hama et al. / International Journal of Mechanical Sciences 50 (2008) 175–192
178
Page 5
vector Durelhas only the third component in the local
frame, xc1and xc2correspond, respectively, to the first and
second components of x0
Hence Eq. (18) can be solved for the unknown quantities Z,
z, and xc3. Moreover, as explained below, the problem is at
most a quartic, and hence its closedform solution exists
[50,51], thus enabling a stable and robust calculation.
The parameters Z and z are obtained through solving the
first and second Eq. (18). Depending on the coefficients in
the first equation, next two cases are considered.
t, which are known quantities.
(a) The case C3
Solving the first Eq. (18) for z yields
16¼0 or C4
16¼0
z ¼ ?C1
1? x1
cþ C1
C1
2Z þ C1
4Z
5Z2
3þ C1
.(19)
Substituting Eq. (19) into the second Eq. (18) gives
C2
6C1
? C1
? C2
Hence the next equation is obtained
1? x1
1? x1
1? x2
cþ C1
cþ C1
cþ C2
2Z þ C1
2Z þ C1
2Z þ C2
5Z2
??2? C1
?¼ 0.
3þ C1
3þ C1
4Z
??
C2
3þ C2
4Z
??
5Z2
5Z2
??þ C1
4Z
??2
?
ð20Þ
MZ4þ NZ3þ OZ2þ PZ þ Q ¼ 0,
where
(21)
M ¼ C1
N ¼ 2C1
4
3C1
?
? C1
?
? C1
þ 2 C1
?
þ C1
?
?2C2
5? C1
4C2
4C1
3C1
?
5C2
5C2
?2C2
4þ C1
4? C1
2þ 2C1
3C2
5
2C1
??2C2
6,
5? C1
5C2
?2C2
?
?2C2
4C2
4
4C1
3þ C1
5? C1
5C2
42C1
5C2
?C1
6,
O ¼ C1
32C1
4? C1
4C2
1? x1
3C1
?2C2
?þ 2 C1
c
?
?
4C2
?
4
3C1
3? C1
1? x1
1? x1
2þ 2C1
1? x1
2C1
?C1
3þ 2C1
5C2
3C2
?
4C2
2þ C2
6,
1? x1
?
c
1? x2
c
?
C1
4
??2
c
6þ C1
4? C1
3C1
?
2
P ¼ ? C1
c
?C1
?C1
2C1
3C2
3? C1
c
??C1
c
4C2
?C1
3
3
?
4C2
1? x2
?2C2
c
1? x1
?þ C1
2C2
6,
?2C2
ð22Þ
Q ¼ ? C1
c
3C2
3þ C1
31? x2
?
1? x1
c
?
6.
Eq. (21) is generally a quartic in Z and is solved
analytically [50,51] in this study. Eq. (21) can
degenerate to be cubic, quadratic or linear depending
on the coefficients and hence the analytical roots can
be obtained in any case. Substituting the root of Z into
Eq. (19) gives z, and then substituting the roots of Z
and z into the third Eq. (18) gives xc3, thus at most four
sets of Z, z, and xc3are attained.
(b) The case C3
The first Eq. (18) becomes a quadratic in Z which
directly gives two roots of Z. Two roots of z are then
derived for each Z by substituting the root of Z into the
1¼ 0 and C4
1¼ 0
second Eq. (18). Consequently, still at most four sets of
Z, z, and xc3are derived.
The roots of Z should fulfill the next condition due to
Eq. (19) when C3
1+C4
1¼ 0 is satisfied
C1
1? x1
cþ C1
2Z þ C1
5Z2¼ 0. (23)
Note that Eq. (21) is equivalent to Eq. (20) and
hence degenerates to Eq. (23) in the case C6
C3
only for C6
the Nagata patch and the physical requirements, the
set of Z, z, and xc3for the intersection point xcshould
satisfy not only Eq. (14) but also the conditions:
26¼0 and
1+C4
1Z ¼ 0. Therefore Eq. (23) needs to be checked
2¼ 0. In addition, due to the definition of
Im Z ð Þ ¼ 0
Therefore, sets of Z, z, and xc3which do not conform
to these conditions are checked off from the following
calculations. Although in most cases only a set of Z, z,
and xc3becomes valid, more sets can remain after testing
Eqs. (14), (23), and (24) when two intersection points lie
within the Nagata patch, as shown in Fig. 2. In such
case, the set for which xc3has the smaller absolute value,
i.e., the intersection point is closer to the assumed sheet
node at the beginning of the increment x0
For instance, in the case of Fig. 2, xc1is adopted as the
tentative intersection point xc. If all the sets of Z, z and
xc3do not satisfy the above conditions, it is considered
that the intersection point xcdoes not lie on the Nagata
patch in question and hence the above calculations are
repeated with other Nagata patches until the tentative
intersection point is attained.
Finally, whether the assumed sheet node x0
the tool or not can be decided from the condition
andIm z ð Þ ¼ 0. (24)
t, is selected.
hpenetrates
n ? xc? xtþDt
0
?
where n is the outward tool normal vector at the
intersection point xc, and can be calculated as
?X0,(25)
n ¼
t1? t2
t1? t2
k k¼
r
r k k,(26)
ARTICLE IN PRESS
x1
x2
x3
x0t
xc1
xc2
x0t+Δt
Δurel
Fig. 2. A case where two intersection points lie on the Nagata patch.
T. Hama et al. / International Journal of Mechanical Sciences 50 (2008) 175–192
179
Page 6
where ? k k denotes the Euclid norm,
t1¼qx
t2¼qx
qZ¼ C2þ C4z þ 2C5Z,
qz¼ C3þ C4Z þ 2C6z,
r ? t1? t2.
The sheet node is considered to penetrate and hence to
contact the tool in question if Eq. (25) is satisfied, while
otherwise this sheet node remains free.
(27)
The abovementioned contact search can be used for the
calculation of rminfor contact, say rc, in the generalized
rminstrategy, which is given as
??
Hence, the penetration of the sheet nodes into the tools can
be prevented by taking into account Eq. (28) when
determining the step size, i.e., the rminvalue. In such case,
furthermore, the contact search for free nodes that is
conducted at the end of the increment can be done by
simply comparing the magnitude of rcwith the rminvalue,
which can be stored for all the candidate contact nodes.
rc¼
xc? xt
Durel
0
??.
??
??
(28)
3.2.3. Contact search for contact nodes
The contact search for a sheet node which is already in
contact with a tool and is sliding on the tool surface is done
by the normal projection of the sheet node at the end of the
increment on the tool surface [29], and hence can be done
by calculating the intersection point xcof the tool normal
vector for the sheet node at the end of the increment with a
Nagata patch. This is schematically shown in Fig. 3, where
the tool normal vector defined at the intersection point xcis
denoted by nt+Dt. On the other hand, according to Eq. (26),
nt+Dtis a nonlinear function of the parameters Z and z of
the initially unknown projection. To avoid this difficulty,
the intersection point xcmay be calculated iteratively, by
taking the ith approximation xcias intersection of the
Nagata patch with the line of direction niand passing
through x0
direction to the patch at xci. The iteration is initialized by
taking n1¼ ntand stops when ni?ni+1is sufficiently close
to 1.
Clearly, at each iteration, we have to solve Eq. (18), in
which the local frame is defined as
t+Dtand then determining ni+1as a normal
~ e3¼ ni;
~ e1¼
~ e3? C6
~ e3? C6
kk;
~ e2¼ ~ e3? ~ e1. (29)
As the time increments in STAMP3D are relatively small
due to the use of the rminstrategy, even the first
approximation xc1and n2is usually sufficiently accurate.
If all sets of Z and z do not satisfy Eqs. (14) and (24) (and
Eq. (23) when C6
that the intersection point xcdoes not lie on the Nagata
patch in question and the above calculations are repeated
with other Nagata patches until the proper intersection
point is obtained.
As described above, the contact search is formulated
based on solving Eq. (18) analytically, thus allowing a
robust and accurate contact search.
2¼ 0 and C3
1+C4
1Z ¼ 0), it is considered
3.2.4. Consistent contact tangent stiffness matrix
Since STAMP3D employs the staticexplicit approach
with the use of rminstrategy, it is essential to follow the
change of tool normal vectors during an increment
explicitly when contact nodes are sliding on curved parts
of tool surfaces. This section describes the discretization of
the second term in the righthand side of Eq. (9), i.e., the
consistent contact tangent stiffness matrix, by taking into
account the Nagata patch.
First, taking an increment of Eq. (26) yields
D^ e3¼ Dn ¼Dr
r k kþ rD
r k k?1
??¼ I ? n ? n
ðÞ ?Dr
r k k,(30)
where I is the unit tensor. From Eq. (27), Dr is given in the
form:
Dr ¼ 2C5DZ þ C4Dz
¼ r1DZ þ r2Dz,
where
ðÞ ? t2þ t1? C4DZ þ 2C6Dz
ðÞ
ð31Þ
r1? 2C5? t2þ t1? C4;
Writing Eq. (31) in a component form with respect to
f^ e1; ^ e2; ^ e3g yields
D^ ri¼ ^ rjiDZj,
where i and j take over 1, 2, 3, and 1, 2, respectively, Z1?Z,
and Z2?z. Substituting Eq. (33) into Eq. (30) gives:
T¼D^ ri
r2? C4? t2þ 2t1? C6. (32)
(33)
D^ ni
T
r k k¼
^ rji
r k kDZj,(34)
where i takes over 1 and 2. Eq. (34) can be written in a
matrix form as
(
r k k
D^ n1
D^ n2
T
T
)
¼
1
^ r11
^ r12
^ r21
^ r22
"#
DZ1
DZ2
()
.(34)0
ARTICLE IN PRESS
x3
x2
x1
x0t+Δt
xc
Δurel
nt+Δt
x0t
Fig. 3. Schematic of the contact search for a sheet node sliding on the
Nagata patch.
T. Hama et al. / International Journal of Mechanical Sciences 50 (2008) 175–192
180
Page 7
Note that D^ nN¼ D^ eN¼ 0 is fulfilled due to the relation
D n ? n
Due to the assumption of the tangent solution, the
relative displacement increment vector Durelof the contact
node is equivalent to the increment of the position vector
on the Nagata patch Dx, thus obtaining
ð Þ ¼ 2n ? Dn ¼ 0.(35)
Durel¼ Dx ¼qx
qZDZ þqx
qzDz ¼ t1DZ þ t2Dz.(36)
Writing in a component and matrix form gives
D^ urel;k
T
¼^tjkDZj, (37)
D^ urel;1
T
D^ urel;2
T
()
¼
^ t11
^t12
^t21
^t22
"#
DZ1
DZ2
()
,(37)0
where k and j take over 1 and 2. The normal component
D^ urel
Solving Eq. (37) for DZj and substituting into Eq. (34)
yields
N¼ 0 is fulfilled by the contact boundary condition.
D^ ni
T¼^ rji^tjk
?
r k k
??1
D^ urel;k
T
,(38)
where i, j, and k take 1 and 2. Writing Eq. (38) in a matrix
form gives
8
:
¼
^k
2122
D^ n1
T
D^ n2
T
<
9
=
;¼
1
r k k
2
^ r11
^ r21
^ r12
^ r22
3
"#
^ t11
^t21
^ t12
^t22
9
"#?1
D^ urel;1
T
D^ urel;2
T
8
:
<
9
;
=
^k
N
11
^k
N
12
N
^k
N
64
75
D^ urel;1
T
D^ urel;2
T
8
:
<=
;,
ð38Þ0
where
2
^k
N
11
N
21
^k
N
12
N
22
^k
^k
4
Eq. (38) is the consistent contact tangent stiffness matrix
taking into account the Nagata patch. Substituting Eq. (38)
into Eq. (9) yields the final form of the discretized virtual
work due to the contact taking into account the Nagata
patch as
Z
X
where i, j, and k takes over 1, 2, and p takes over 1, 2,
and 3. dij is the Kronecker delta. Writing in a matrix
3
5?
1
r k k
^ r11
^ r12
^ r21
^ r22
"#
^t11
^t12
^ t21
^ t22
"#?1
.(39)
S3
_f ? dvdS
??
Dt
¼
l
dl^ vp aDl^ urel
Ndp3þl^f
c
N
l^ rji
l^tjk
????
???1
lr
Dl^ urel;k
T
dpi
!
,
ð40Þ
form gives
Z
2
S3
_f ? dvdS
??
Dt ¼
X
N
12
l
dl^ v1
dl^ v2
dl^ v3
n
0
o
?
l ^f
c
N
l ^k
N
11
l ^f
c
N
l ^k
l ^f
c
N
l ^k
N
21
l ^f
c
N
l ^k
N
22
0
00
a
6664
3
7775
Dl^ urel;1
T
Dl^ urel;2
T
Dl^ urel
N
8
>
>
>
>
>
>
:
<
9
>
>
>
>
>
>
;
=
.
ð41Þ
The consistent contact tangent stiffness matrix is the
function of the relative displacement increment of sheet
node l and Dlurel, thus incorporated into the tangent
stiffness matrix in the tangent stiffness equation.
The formulation stated above cannot completely follow
the change of tool normal vector since this is only the first
approximation, while the Nagata patch is the quadratic
parametric surface. Hence, the canceling algorithm de
scribed in Section 2.2 is simultaneously employed such that
the remaining nonequilibrated forces due to the highly
nonlinear phenomena are cancelled explicitly.
Although the contact treatment taking into account the
Nagata patch stated above is formulated in the framework
of the tangent solution in which iterations for nodal
equilibrium are not employed, the key techniques are of
course applicable to other finiteelement codes without loss
of generality.
3.3. Contact treatment used in conventional STAMP3D
Contact
STAMP3D are described in the original papers [32,33,46]
in detail. Only their outlines are described briefly here.
treatments employed inconventional
3.3.1. Polyhedral approximation of surface
In conventional STAMP3D, the tool surfaces approxi
mated by a set of planar triangular patches are directly
used. A unit tool normal vector nkat tool node k is
uniquely defined by averaging the tool normal vectors
defined on all adjacent elements as
P
m
where nk
mis the unit normal vector at node k as the cross
product of two tangential vectors to the adjacent element
m. Smis the weighting factor, which is taken equal to the
area of the element m. Thereafter, a tool normal vector at
an arbitrary point is defined by interpolation with use of
shape functions3Nkfor a 3node triangular element as
P3
nk¼
mSmnk
mSmnk
m
P
????,(42)
n ¼
k¼1
k¼13Nknk
3Nknk
P3
??????
¼
V
V
k k,(43)
where
V ?
X
3
k¼1
3Nknk.(44)
ARTICLE IN PRESS
T. Hama et al. / International Journal of Mechanical Sciences 50 (2008) 175–192
181
Page 8
3Nk, k ¼ 1, 2, 3 are linear functions of the area coordinates
of the tool element. It should be noted that, using the
above definitions, the normal vectors change continuously
over the tool surface, while they are not necessarily
consistent with the shape of the tool surface.
3.3.2. Contact search algorithm
The contact search is carried out between sheet nodes and
planar triangular patches using the similar procedures
explained in Sections 3.2.2 and 3.2.3 both for the free and
contact nodes, thus coming to calculating the intersection
point of a projection vector from a sheet node with a planar
triangular patch. Clearly, the intersection point can be
calculated analytically as well as the Nagata patch model.
3.3.3. Consistent contact tangent stiffness matrix
Discretizing the second term in the righthand side of Eq. (9)
using Eq. (43) in the similar way as that in Section 3.2.4, the
discretized virtual work due to the contact is derived as
Z
X
where
X
X
i, j, k takes over 1, 2, and p takes over 1, 2, 3. Writing in a
matrix form gives
Z
l
l ^f
N
6664
S3
_f ? dvdS
??
Dt
¼
l
dl^ vp aDl^ urel
Ndp3þl^f
c
N
l ^Aij
l^Bkj
??
???1
lV
??
Dl^ urel;k
T
dpi
!
, ð45Þ
^Aij¼
3
h¼1
3
q3Nh
qxj
^ nh
i,(46)
^Bkj¼
h¼1
q3Nh
qxj
^ xh
k,(47)
S3
_f ? dvdS
??
Dt ¼
X
2
dl^ v1
dl^ v2
dl^ v3
no
0
?
c
l ^k
C
11
l ^f
c
N
l ^k
C
12
l ^f
c
N
l ^k
C
21
l ^f
c
N
l ^k
C
22
0
00
a
3
7775
Dl^ urel;1
T
Dl^ urel;2
T
Dl^ urel
N
8
>
>
>
>
>
>
:
<
9
>
ð48Þ
>
>
>
>
>
;
=
,
where
^k
C
11
^k
C
12
^k
C
21
^k
C
22
2
64
3
75 ¼
1
V
k k
P
P
q3Nh
qz1
3
h¼1
3
q3Nh
qz1
^ nh
1
P
P
q3Nh
qz2
3
h¼1
3
q3Nh
qz2
^ nh
1
h¼1
q3Nh
qz1
^ nh
2
h¼1
q3Nh
qz2
^ nh
2
2
66664
P
P
3
77775
?1
?
3
h¼1
3
^ xh
1
P
P
3
h¼1
3
^ xh
1
h¼1
q3Nh
qz1
^ xh
2
h¼1
q3Nh
qz2
^ xh
2
2
66664
3
77775
.
ð49Þ
4. Simulation of a twodimensional draw bending process of
a high strength steel
4.1. Simulation conditions
In order to validate the proposed contact treatment and
to examine the effect of tool modeling accuracy on
springback analysis, simulations of a twodimensional
draw bending process [47] of a high strength steel sheet
are carried out. The mechanical properties of the sheet
material are shown in Table 1. In order to emphasize the
effects of contact modeling, the anisotropy of sheet is not
taken into consideration. The geometries of tools employed
in the simulation are shown in Fig. 4 [47]. The size of the
blank is 5mm wide, 200mm long, and 1.0mm thick. Due
to the symmetry of the process, only the half part is
modeled. The plane strain condition is assumed in the
width direction, to which corresponding boundary condi
tions are given. Fournode degenerated shell elements with
assumedstrain integration [5,43] are being used and the
number of elements are 1 and 100 along the width and
longitudinal directions, respectively. Sheet discretization is
one of the important factors for springback simulations.
From our experiences, it would be rather worse to make
the element size be smaller than the shell thickness. The
sheet mesh in the longitudinal direction described above is
therefore chosen although the element size may be
considered minimal to describe the punch and die radii.
ARTICLE IN PRESS
Table 1
Material properties used in the simulationsa
E (GPa)
n
sy(MPa)
F (MPa)
n
e0
200
0.3
760
1600
0.12
0.0021
aThe true stress–true plastic strain curve is approximated by the Swift
law s ¼ F(e0+ep)n.
x
z
27.6
R3
R5
30
Blank holder
Die
Punch
Fig. 4. Schematic of tool geometries in mm.
T. Hama et al. / International Journal of Mechanical Sciences 50 (2008) 175–192
182
Page 9
On the other hand, we adopt the sheet mesh with the plane
strain constraint in the width direction in order to reduce
the effect of the difference in the tool modeling accuracy in
the width direction on the simulated results as much as
possible, since the deformation in the present twodimen
sional draw bending process should be uniform in the
width direction except for the vicinity of the sheet edge.
The tool modeling accuracy in the present tool models will
be described in the next section in detail. Fifteen Gauss
integration points are introduced through thickness.
Friction is not taken into consideration, while, as will be
described below in detail, stretching forces equivalent to
the friction force due to the blank holding force are given
to the sheet prior to the drawing process.
Calculation procedures adopted in the present draw
bending simulation consist of the following three steps.
(i) Stretching forces are applied on the sheet edge until the
tensile stress s reaches a prescribed value, which is
varied from around 1% to 40% of the initial yield
stress sy.
(ii) With keeping the initial stretching forces, the sheet is
drawn up to the punch stroke of 70mm.
(iii) Finally, nodal forces opposite to the internal forces are
applied until the latter are cancelled. (This process
corresponds to springback calculation.)
Comparisons of the simulation results with the experi
mental ones are not conducted in this paper from the
following reasons: (a) simple simulation conditions of
friction and constitutive models are chosen in this study in
order to clarify the effect of tool model accuracy on the
results and (b) the scattering in experiments may cover
the difference observed in the simulation results. Instead,
the simulation results are compared with the analytical
ones based on the elastoplastic incremental strain theory
proposed by Kuwabara et al. [52,53]. This analysis models
a metal sheet subjected to bending–unbending deformation
under tension, which corresponds to the deformation at the
die shoulder in the present process. It has been clarified
that the analytical results agree rather well with the
experimental ones. It is considered that, if the simulations
are properly carried out, the simulation results will agree
with the analytical ones unless contact between the sheet
and the sidewall of the punch takes place.
This forming process is rather simple, but involves the
bending–unbending deformation, a large influence of
stretching force at the flange, and sometimes the socalled
overrun phenomenon. Hence it is difficult to predict a
deformed profile after springback accurately.
4.2. Tool modeling accuracy
Since the sheet slides continuously on the die shoulder
during the process, the modeling accuracy at the die
shoulder may largely affect the simulation results. There
fore, two different discretized die models, dies A and B
(Fig. 5), are used. The die shoulder is finely discretized in
die A, and somewhat roughly in die B. Three types of tool
modeling are employed in this study. The first approach is
a tool modeling used in STAMP3D, which is subsequently
called the conventional polyhedral model (see Section 3.3).
The second approach employs the Nagata patch for the
tool surfaces. In the third approach, the polyhedral
surfaces are directly employed for the shape description,
while the tool normal vectors that are consistent with the
original CAD surfaces are given at the tool nodes instead
of the averaged normal vectors defined in Eq. (42). A tool
normal vector at an arbitrary point is defined by the
interpolation (Eq. (43)) as in the conventional polyhedral
model. This tool modeling is subsequently called the new
polyhedral model. These three approaches are employed
both for dies A and B, thus attaining six die models. As for
the punch and the blankholder, finely discretized models
as they are usually employed in the conventional poly
hedral model are used (Fig. 6). In the calculations, all the
tools, i.e., the punch, the die, and the blank holder, are
represented by a same tool modeling.
ARTICLE IN PRESS
10
x, ex
y, ey
D
C
z, ez
G
G
H
H
10
135
100
D
C
G
G
H
H
6.25
100
100
135
Fig. 5. Die models employed in the simulations in mm. The dotted lines
GG and HH correspond to the boundaries of the shoulder and straight
parts in the exact die profile. The points C and D are on the lines GG and
HH, respectively. (a) Die A and (b) die B.
T. Hama et al. / International Journal of Mechanical Sciences 50 (2008) 175–192
183
Page 10
The shape and normal vector error distributions at the
die shoulder are compared between dies A and B, and
between the Nagata patch, conventional polyhedral, and
new polyhedral models. The shape and normal vector
errors dsand dnat point p are respectively defined as
ds¼ rds? x ? o
?
where x is the position vector of point p, rds¼ 5 [mm] is the
die shoulder radius, and o is the position vector of the
center of the die shoulder radius. n ¼ x ? o
the die shoulder, while n ¼ ?ez at the upper part and
n ¼ ?exat the sidewall, V is the unit tool normal vector at
point p calculated on the die model, and t is the unit vector
tangent to the exact tool surface and satisfies both the
relations n?t ¼ eyand n?t ¼ 0, where ex, ey, and ezare the
base vectors shown in Fig. 5.
Fig. 7 shows the shape error distributions along line CD
shown in Fig. 5. Note that the shape description for the
new polyhedral and conventional polyhedral models are
the same. Line CD is chosen since the sheet nodes slide on
this line. In the polyhedral model, the negative shape errors
distribute periodically depending on the tool mesh both in
dies A and B. The shape error is smaller around the
element boundary, and larger around the center of the ele
ment. Although somewhat periodical errors exist in the
Nagata patch model as well, they are too small to be visible
in Fig. 7. Clearly, the Nagata patch model yields a much
more accurate shape than the polyhedral model both in
dies A and B. Comparing die A with die B, die A yields a
more precise shape than die B in the polyhedral model,
while die B is as accurate as die A in the Nagata patch
model except for the ends of the die shoulder.
Fig. 8 shows the normal vector error distributions along
line CD. The level of accuracy for the four models, the
ð Þ ? n
ðÞ=rds? 100
?
½%?,(50)
dn¼ sgn t ? V ? n
ðÞ
V ? n
k k ? 100
½%?,(51)
ðÞ= x ? o
kk at
Nagata patch models for dies A and B and the new
polyhedral models for dies A and B, are almost the same
except at the ends of the die shoulder. The new polyhedral
models yield rather better distributions at the die shoulders
as shown in the emphasized figures. On the other hand, the
conventional polyhedral model yields much larger errors
than the others both in dies A and B. The conventional
polyhedral model for die A yields a negative normal vector
error distribution in the whole region, while the normal
vector errors for die B change the signs and their
magnitude is larger than that of die A. These results show
that the interpolation of normal vectors (Eq. (43))
practically gives an accurate distribution as that of the
Nagata patch when accurate normal vectors are defined at
the tool nodes, and that the averaging of normal vectors
(Eq. (42)) gives a worse result compared to the others in the
present models.
The large errors are shown at the ends of the die
shoulder both in the shape and in the normal vector in all
the models. These errors are due to the fact that the
original CAD data is represented by one surface so that the
element boundary is not on the boundary of the straight
and curved parts both in dies A and B. Therefore, to reduce
such large errors, CAD data with explicit specification of
ARTICLE IN PRESS
Punch
Blank holder
Die (die A)
Fig. 6. Discretized punch and blank holder models employed in the
simulations.
0.6
0.4
0.2
0
0.500.51 1.5
0.6
0.4
0.2
0
0.50 0.5 1.5
Shape error ?s%
Dimensionless length from C l
Nagata patch
Polyhedral
Shape error ?s%
Dimensionless length from C l
Nagata patch
Polyhedral
Die shoulderUpper part Sidewall
Die shoulderUpper part
CD
CD
1
Sidewall
Fig. 7. Shape error distributions at die shoulders. (a) Die A and (b) die B.
T. Hama et al. / International Journal of Mechanical Sciences 50 (2008) 175–192
184
Page 11
curvature discontinuities should be used not only for
the polyhedral models but also for the Nagata patch
models.
4.3. Results and discussion
4.3.1. Simulation results for die A
Fig. 9(a) shows the deformed profiles after springback
for die A for the ratio of initial tensile stress and initial
yield stress s/sy¼ 0.14. The simulation result for the new
polyhedral model agrees well with that of the Nagata patch
model, while that of the conventional polyhedral model
differs from the other two results and yields much smaller
springback at the sidewall. The simulation is apparently
affected by the tool modeling.
To examine the simulation results more in detail, the
relationship between the sidewall curvature after spring
back and the initial tensile stress is shown in Fig. 9(b). The
sidewall curvature is calculated as follows. Fig. 10 shows an
example of the curvature distribution along the long
itudinal direction of the sheet after springback by
approximating three nodes for every three nodes with a
circle. Since the sidewall curvature is fairly uniform as
shown in Fig. 10, an average curvature of a region
circumscribed by a dotted square is adopted in Fig. 9(b).
The deformed profiles shown in Fig. 9(a) correspond to the
results for s/sy¼ 0.14 in Fig. 9(b). The peak curvature
occurs at around s/sy¼ 0.14 for the Nagata patch and new
polyhedral models, and their results agree fairly well with
each other. On the other hand, the peak curvature occurs
at around s/sy¼ 0.36 for the conventional polyhedral
model, and the result differs from the other two results in
the low tensile stress region. Generally speaking, the
curvature decreases as the initial tensile stress decreases
in the low tensile stress region since the reverse bending due
to the overrun phenomenon increases as the initial tensile
stress decreases [37,54]. Therefore, Fig. 9(b) indicates that,
in the conventional polyhedral model, the reverse bending
starts occurring at higher initial tensile stress than in the
other two models.
To confirm this rationale, the variations of the distance
from the sidewall of the punch to the upper assumed
surface of the sheet at the punch stroke of 70mm for
s/sy¼ 0.04, 0.14, and 0.26 are examined in Fig. 11. Two
points a and b are chosen on the sheet and the longitudinal
stress distributions through the thickness at the punch
stroke of 70mm for s/sy¼ 0.04, 0.14, and 0.26 are at the
same time examined in Figs. 12–14. The analytical stress
distributions [52,53] are also shown in the figures. It is
noted that point a is that immediately after experiencing
the unbending. When using the conventional polyhedral
model, the sheet already contacts the sidewall of the punch
for s/sy¼ 0.26, inducing the apparent reverse bending for
all the conditions (Fig. 11(a)). Consequently, although the
simulated stresses are in good agreement with the
analytical ones at point a (z ¼ 8.9mm) (Figs. 12(a), 13(a),
ARTICLE IN PRESS
10
5
0
5
10
0.50 0.511.5
10
5
0
5
10
0.50 0.51 1.5
0.04
0.02
0
0.02
0.04
00.51
Normal vector error ?n%
Dimensionless length from C l
Dimensionless length from C l
Nagata patch
Conventional polyhedral
Normal vector error ?n%
New polyhedral
Nagata patch
Conventional polyhedral
New polyhedral
0.04
0.02
0
0.02
0.04
00.51
Normal vector error ?n%
Dimensionless arc length from C toD l
Nagata patch
New polyhedral
Normal vector error ?n%
Dimensionless arc length from C to D l
Nagata patch
New polyhedral
Die shoulderSidewallUpper part
Die shoulder Sidewall Upper part
CD
CD
CD
CD
Fig. 8. Normal vector error distributions at die shoulders. (a) Die A and (b) die B. (Figures on the right side are emphasized figures for the ranges of
?0.04odno0.04 and 0olo1.0. Only the results of the new polyhedral and Nagata patch models are shown.)
T. Hama et al. / International Journal of Mechanical Sciences 50 (2008) 175–192
185
Page 12
14(a)), they decrease at the sidewall (point b, z ¼ 48.3mm)
(Figs. 12(b), 13(b), 14(b)) for all the conditions. Since the
reverse bending increases as the initial tensile stress
decreases, the simulated stress also decreases at the sidewall
as the initial tensile stress decreases.
When using the Nagata patch and new polyhedral models,
the reverse bending is not observed for s/sy¼ 0.26, while is
observed for s/sy¼ 0.14 and 0.04 (Fig. 11(b), (c)). Conse
quently, the simulated stresses remain unchanged even at the
sidewall (point b) for s/sy¼ 0.26 (Fig. 14(b)), while decrease
largely for s/sy¼ 0.04 (Fig. 12(b)). Since the reverse bending
shown for s/sy¼ 0.14 is minute, the simulated stresses
hardly decrease at the sidewall (Fig. 13(b)). The occurrence
timing of the reverse bending in the conventional polyhedral
model is clearly different from the others, resulting in the
different stress distributions at the sidewall, and ultimately in
the smaller curvature after springback for s/syo0.36, as
shown in Fig. 9.
4.3.2. Simulation results for die B
Fig. 15(a) shows the deformed profiles after springback
for die B for s/sy¼ 0.14. Unlike the case of die A
(Fig. 9(a)), the simulation result for the conventional
polyhedral model agrees well with that of the Nagata patch
ARTICLE IN PRESS
10
10
Nagata patch
Conventional
polyhedral
New polyhedral
Profile before springback
0
5
10
15
20
0 0.20.40.6
Ratio of initial tensile stress and yield stress ?/?y
Sidewall Curvature ?/m1
: Nagata patch
: Conventional polyhedral
: New polyhedral
Fig. 9. Simulated results for die A. (a) Deformed profiles after springback
for s/sy¼ 0.14 in mm. (b) Relationship between the sidewall curvature
after springback and the initial tensile stress.
0
0.04
0.08
0.12
0.16
0.2
0.24
020406080100
Distance from the symmetric plane l /mm
Curvature ?/m1
Sidewall Flange
Die shoulder
Punch shoulder
Punch bottom
×1000
Fig. 10. Distribution of curvature along the deformed profile after
springback.
Sheet
z
Punch
d
0
0.05
0.1
0.15
0.2
0.25
02030 40506070
0
0.05
0.1
0.15
0.2
0.25
010 20 3040 50 6070
0
0.05
0.1
0.15
0.25
0102030 40506070
z/mm
0.04
0.26
?/?y = 0.14
d/mm
Sidewall Punch bottomFlange
z/mm
?/?y = 0.04
0.26
0.14
d/mm
SidewallPunch bottom Flange
z/mm
?/?y = 0.04
0.14
0.26
d/mm
SidewallPunch bottom Flange
0.2
10
Fig. 11. Variation of distance from the sidewall of the punch to the sheet
at the punch stroke of 70mm for die A. (a) Conventional polyhedral
model, (b) new polyhedral model, and (c) Nagata patch model.
T. Hama et al. / International Journal of Mechanical Sciences 50 (2008) 175–192
186
Page 13
model, while that of the new polyhedral model differs from
the others and yields much smaller springback at the
sidewall. The relationship between the sidewall curvature
after springback and the initial tensile stress is examined in
Fig. 15(b). The result for the conventional polyhedral
model tolerably agrees with that of the Nagata patch
model, while that of the new polyhedral model largely
differs from the other two results and the peak curvature
occurs at higher initial tensile stress.
As is the case with the conventional polyhedral model for
die A, this difference is due to the fact that the reverse bending
starts occurring at higher initial tensile stress for the new
polyhedral model. This is supported by Figs. 16–19, which
show the deformed profiles at the sidewall and the longitudinal
stress distributions through the thickness at points a and b at
the punch stroke of 70mm for s/sy¼ 0.04, 0.14, and 0.26,
respectively. For the new polyhedral model, the reverse
bending is not induced for s/sy¼ 0.26 but for s/sy¼ 0.04
and 0.14 (Fig. 16(b)). Consequently, the simulated stress at the
sidewall (point b, z ¼ 48.3mm) remains unchanged from point
a (z ¼ 8.9mm) for s/sy¼ 0.26 (Fig. 19), while decreases for
s/sy¼ 0.04 and 0.14 (Figs. 17 and 18). For the conventional
polyhedral and Nagata patch models, the reverse bending
and the decrease in the simulated stress are clearly induced
only for s/sy¼ 0.04, and these results fairly agree with each
other.
ARTICLE IN PRESS
0.5
0
0.5
20002000
0.5
0
0.5
200002000
Longitudinal stress ?l/MPa
Distance from the
midsurface h /mm
Longitudinal stress ?l/MPa
Distance from the
midsurface h /mm
a
b
0
Fig. 12. Longitudinal stress distributions at points a and b for s/sy¼ 0.04 for die A. The solid line is the analytical result [52,53]. J, m, and ~ are the
results for the Nagata patch, conventional polyhedral, and new polyhedral models, respectively. (a) Point a and (b) point b.
0.5
0
0.5
20002000
0.5
0
0.5
200002000
Longitudinal stress ?l/MPa
Distance from the
midsurface h /mm
Longitudinal stress ?l/MPa
Distance from the
midsurface h /mm
a
b
0
Fig. 13. Longitudinal stress distributions at points a and b for s/sy¼ 0.14 for die A. The solid line is the analytical result [52,53]. J, m, and ~ are the
results for the Nagata patch, conventional polyhedral, and new polyhedral models, respectively. (a) Point a and (b) point b.
0.5
0
0.5
20002000
0.5
0
0.5
20000 2000
Longitudinal stress ?l/MPa
Distance from the
midsurface h /mm
Longitudinal stress ?l/MPa
Distance from the
midsurface h /mm
a
b
0
Fig. 14. Longitudinal stress distributions at points a and b for s/sy¼ 0.26 for die A. The solid line is the analytical result [52,53]. J, m, and ~ are the
results for the Nagata patch, conventional polyhedral, and new polyhedral models, respectively. (a) Point a and (b) point b.
T. Hama et al. / International Journal of Mechanical Sciences 50 (2008) 175–192
187
Page 14
4.3.3. Discussion
We recall that the die shoulder is finely discretized in die
A, and somewhat roughly in die B, as shown in Fig. 5. The
results described in Sections 4.3.1 and 4.3.2 clearly show
that the simulation results including the deformed profile
after springback are strongly affected by the tool modeling.
It is interesting that both the new and conventional
polyhedral models show opposite tendencies between dies
A and B. The occurrence of the reverse bending is in
general governed by forming conditions, such as die radius
and blank holding force which corresponds to the initial
stretching force in the present simulation. However, the
differences shown in the present results can be rather
explained by the tool modeling errors shown in Figs. 7
and 8. Fig. 20 shows the variations of the distance from the
exact die shoulder to the sheet nodes at the punch stroke of
70mm for dies A and B for s/sy¼ 0.14. Note that negative
distance indicates that the sheet node penetrates the exact
die shoulder. As for die A, the shape errors for the
conventional polyhedral model are same as those of the
new polyhedral model, while the normal vector errors are
not only larger than those of the new polyhedral model
but also negative in the whole region (Fig. 8(a)). Due to
these large negative errors, when using the conventional
polyhedral model, the contact nodes slide in the direction
releasing from the die shoulder, and hence the sheet does
not fit the die shoulder firmly, as shown in Fig. 20(a).
Therefore it becomes easier to induce the reverse bending.
On the other hand, when using the new polyhedral model,
the contact nodes slide almost exactly tangent to the die
shoulder in each increment since the normal vectors are
much more accurate, thus fitting the die shoulder well.
Therefore the reverse bending is more difficult to be
induced.
As for die B, in the new polyhedral model, the normal
vectors are as accurate as that of die A (Fig. 8), while the
shape errors are larger than those of die A (Fig. 7). Due to
these large shape errors, the sheet slightly penetrates the
exact die shoulder as shown in Fig. 20(b), thus being
geometrically easier to release from the die shoulder and
ultimately to induce the reverse bending. In the conven
tional polyhedral model, although both the shape and
normal vector errors are much larger than those of die A,
the effect of the normal vector errors is dominant to the
deformation due to their magnitude. Since the normal
ARTICLE IN PRESS
10
10
New polyhedral
Conventional
polyhedral
Nagata patch
Profile before springback
0
5
10
15
20
00.20.40.6
Ratio of initial tensile stress and yield stress ?/?y
Sidewall Curvature ?/m1
: Nagata patch
: Conventional polyhedral
: New polyhedral
Fig. 15. Simulated results for die B. (a) Deformed profiles after spring
back for s/sy¼ 0.14 in mm and (b) Relationship between the sidewall
curvature after springback and the initial tensile stress.
Sheet
z
0
0.05
0.1
0.15
0.2
0.25
0 203040506070
0
0.05
0.1
0.15
0.2
0.25
010 2030 40506070
0
0.05
0.1
0.15
0.25
0 1020 304050 6070
z/mm
d/mm
SidewallPunch bottomFlange
z/mm
?/?y = 0.04
d/mm
SidewallPunch bottomFlange
z/mm
d/mm
SidewallPunch bottomFlange
0.2
10
?/?y = 0.04
0.26
0.14
0.14
0.26
0.26
0.14
?/?y = 0.04
Punch
d
Fig. 16. Variation of distance from the sidewall of the punch to the sheet
at the punch stroke of 70mm for die B. (a) Conventional polyhedral
model, (b) new polyhedral model, and (c) Nagata patch model.
T. Hama et al. / International Journal of Mechanical Sciences 50 (2008) 175–192
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Page 15
vector errors take positive and negative values almost
equally, the effect of the normal vector error may be
cancelled by chance, thus fitting the die shoulder well
(Fig. 20(b)). Therefore the reverse bending is more difficult
to be induced.
The above mechanisms show that not only the shape but
also the normal vectors apparently play quite important
roles in the simulation accuracy of springback, and hence it
is clear that the use of polyhedral model is not always
acceptable since this cannot escape from the possibility of
affecting the simulation result in any manner.
The simulation results for the Nagata patch model yield
quite different tendencies from those of the polyhedral
models. As shown in Figs. 9, 11(c), 15, and 16(c), the
results for dies A and B show similar tendencies. This
correspondence can also be seen in Fig. 20, which shows
that the sheet fits the die shoulder well both for dies A and
B. Fig. 21 shows the comparisons of the relationship
between the sidewall curvature after springback and the
initial tensile stress between dies A and B for each tool
modeling. The large differences are observed both for the
conventional and new polyhedral models as explained
above. On the other hand, when using the Nagata patch
model, the two results fairly agree with each other,
although a slight difference is observed at the low initial
tensile stress region. These results show the level of
accuracy of the shape and the normal vectors are adequate
in the Nagata patch model both for dies A and B for the
ARTICLE IN PRESS
0.5
0
0.5
20002000
0.5
0
0.5
200002000
Longitudinal stress ?l/MPa
Distance from the
midsurface h /mm
Longitudinal stress ?l/MPa
Distance from the
midsurface h /mm
a
b
0
Fig. 18. Longitudinal stress distributions at points a and b for s/sy¼ 0.14 for die B. The solid line is the analytical result [52,53]. J, m, and ~ are the
results for the Nagata patch, conventional polyhedral, and new polyhedral models, respectively. (a) Point a and (b) point b.
0.5
0
0.5
20002000
0.5
0
0.5
20000 2000
Longitudinal stress ?l/MPa
Distance from the
midsurface h /mm
Longitudinal stress ?l/MPa
Distance from the
midsurface h /mm
a
b
0
Fig. 19. Longitudinal stress distributions at points a and b for s/sy¼ 0.26 for die B. The solid line is the analytical result [52,53]. J, m, and ~ are the
results for the Nagata patch, conventional polyhedral, and new polyhedral models, respectively. (a) Point a and (b) point b.
0.5
0
0.5
20002000
0.5
0
0.5
200002000
Longitudinal stress ?l/MPa
Distance from the
midsurface h /mm
Longitudinal stress ?l/MPa
Distance from the
midsurface h /mm
a
b
0
Fig. 17. Longitudinal stress distributions at points a and b for s/sy¼ 0.04 for die B. The solid line is the analytical result [52,53]. J, m, and ~ are the
results for the Nagata patch, conventional polyhedral, and new polyhedral models, respectively. (a) Point a and (b) point b.
T. Hama et al. / International Journal of Mechanical Sciences 50 (2008) 175–192
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