=Paper=
{{Paper
|id=Vol-1638/Paper69
|storemode=property
|title=Effect of anisotropic yield criterion on springback in plane strain pure bending
|pdfUrl=https://ceur-ws.org/Vol-1638/Paper69.pdf
|volume=Vol-1638
|authors=Fedr V. Grechnikov,Yaroslav A. Erisov,Sergei E. Alexandrov
}}
==Effect of anisotropic yield criterion on springback in plane strain pure bending ==
https://ceur-ws.org/Vol-1638/Paper69.pdf
Mathematical Modeling
EFFECT OF ANISOTROPIC YIELD CRITERION ON
THE SPRINGBACK IN PLANE STRAIN PURE
BENDING
F.V. Grechnikov1,2, Ya.A. Erisov1, S.E. Alexandrov3
1
Samara National Research University, Samara, Russia
2
Samara Scientific Center of Russian Academy of Sciences, Samara, Russia
3
Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, Russia
Abstract. Plastic anisotropy is one of the multiple material representations sig-
nificantly affecting the springback simulations. The present paper studies the in-
fluence of the plastic anisotropy over the springback in plane strain pure bend-
ing by means of the exact semi-analytical solution. The yield criterion and the
constitutive equations for the orthotropic material with consideration of the
crystal lattice constants and parameters of the crystallographic texture are used
taking into account the anisotropy.
Keywords: anisotropy, crystallographic orientation, yield criterion, springback,
bending, plane strain, elastic/plastic solution.
Citation: Grechnikov FV, Erisov YaA, Alexandrov SE. Effect of anisotropic
yield criterion on the Springback in plane strain pure bending. CEUR Work-
shop Proceedings, 2016; 1638: 569-577. DOI: 10.18287/1613-0073-2016-1638-
569-577
1 Introduction
As it is evident from the production analysis of the Russian plants the metal loss in sheet
metal forming processes only amounts up to 45-50%: metal utilization factor in aircraft
industry is about 40%, in engine industry - 30% and in automotive industry - 50%. Pre-
sumably, waste is much higher due to the collateral loss (about 10-15%). In a greater
degree, these losses arise from the undesirable anisotropy of the blank properties.
For most materials the anisotropy of physical and mechanical properties is a rule ra-
ther than an exception [1]. Generally, anisotropy, which stems from the crystal struc-
ture of materials, determines processing and service characteristics of the products. As
to metal forming processes, the undesirable anisotropy not only distorts the shapes
and sizes of the parts, but also limits the formability, causing the excessive thinning of
the material [2]. Therefore, in order to compensate thinning, it is necessary to increase
the initial sizes of the blanks, which in turn results in the metal loss, the weight
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growth of the construction, the technological cycle extension, and, consequently, in
the numerous additional expenses (equipment, areas, power resources, workers, etc).
However, as it follows from the numerous researches [3-6], the influence of anisotro-
py has more advantages rather than disadvantages. For example, the considerable
achievements in material science of the electric steels (e.g. increasing magnetic induc-
tion and capacitance, cutting electric power losses) became possible mainly due to
developing efficient anisotropy of physical properties [7-8].
From the above it appears, firstly, the instant necessity of the anisotropy impact as-
sessing in processing and service characteristics of products; secondly, the necessity
of developing special methods for obtaining and efficient application of anisotropy.
Unfortunately, the imperfection of the process design techniques does not allow con-
sidering anisotropy of the blank properties [9-10] and, as a result, makes the excessive
metal consumption inevitable at the production stage.
A comprehensive overview on springback that follows a sheet forming process when
the forming loads are removed from the workpiece has been provided in [11]. It is
emphasized that plastic anisotropy is one of the various material representations that
significantly affects the springback simulations. In the present paper, the effect of the
springback plastic anisotropy in the plane strain pure bending is studied by means of
the exact semi-analytical solution.
2 Theory of bending at large strain
The theory of the elastic/plastic plane strain bending of an incompressible anisotropic
sheet obeying the quadratic Hill yield criterion [12] has been presented in the article
[13]. Ultimately, the main results of this work are represented in this section. The
solution is based on the following mapping between the Eulerian Cartesian coordinate
system (x, y) and the Lagrangian coordinate system ( , )
x s s y s
cos 2a , sin 2a , (1)
H a a 2 a H a a2
where H is the initial thickness of the sheet, s is an arbitrary function of a , which is
a function of the time, t , and a 0 at t 0 . It follows from Eq. (1) that x H and
y H at a 0 if
s 1 4 at a 0. (2)
It can be verified by the inspection through the application of l’Hospital’s rule to
Eq. (1), with the use of Eq. (2), as a 0 . Eq. (1) and (2) a transformation description
of the rectangle defined at the initial instant a 0 , by the equations x H , x 0
and y L (or, in the Lagrangian coordinates, by the equations 1 , 0 and
L H ) into the shape determined by the two circular arcs, AD and CB, and the
two straight lines, AD and CB (Fig. 1).
It is convenient to introduce a moving cylindrical coordinate system ( r, ) by the
equations of transformation
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x s r cos y r sin
and . (3)
H a H H H
Fig. 1. Geometry of the process - notation
The origin of this coordinate system is located at x H s a and y 0 . The lines
CB and DA are determined by the equations 0 , and the lines AB and CD by the
equations r rAB and r rCD , respectively (Fig. 1). 0 , rAB , and rCD are the func-
tions of a . Using Eq. (1), it is also possible to get the following relations
rAB s r s 1 2aL h s sa
, CD , 0 , , (4)
H a H 2 a H H a
a
where h is the current thickness.
In the case of the elastic/plastic bending, there are two plastic regions and one elastic
region. Let 1 a and 2 a be the elastic-plastic boundaries moving from
0 and 1 , respectively. Then, the distribution of the stresses is given by
a s as
ln , ln 2 (5)
T s T s
in the region 1 a 0 ,
a s sa
ln , ln 2 (6)
T s a T a s
in the region 1 2 a and
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1 2 p
ln 4 a s ln 4s ,
T p 4
(7)
1 p 4
ln 2 4 a s ln 4s ln 4 a s
T p 4 p
in the region 2 a 1 a . Here
2a 1 4a 2 exp p 2 4 s exp p 2 4 s
s , 1 a , 2 a ,
4 4a 4a
X 2Y 2 Z 2T (8)
T , p ,
2 G
2 X 2Y 2 Z 2 Y 2 Z 2 Y 4 Z 4 X 4 Y 2 Z 2
where X , Y and Z are the tensile yield stresses in the - , - and z - directions,
respectively, and G is the shear modulus of elasticity.
The bending moment per unit length is defined by
rAB
M rdr.
rCD
Using Eq. (1) and (3), this equation can be transformed to
1
0
2M
a T
m d , (9)
TH 2 1
where m is the dimensionless bending moment.
In the case of the purely elastic unloading, the stress increments are
C1
4 R0 R1
2 2
r R0 2
ln
p R12 ln R1 R0 R0 r
1 ,
T
4 R0 R1
2 2
r R02
C1 ln 1 1, (10)
T p R12 ln R1 R0 R0 r 2
1
2
2
mf p H R12 R02 1
R1
C1 4 ln .
2 R0 R 2 R 2 ln R R
1 0 1 0 R0
Here m f , R0 and R1 are the values of m , rCD and rAB at the end of loading. Also,
r sf
, (11)
H af a 2f
where a f and s f are the values of a and s at the end of loading. Using Eq. (7), in
which a and s should be replaced with a f and s f from Eq. (11), the variation of
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and with r H at the end of loading is determined in parametric form with
being the parameter. Then, the distribution of the residual stresses are found from
res
f
,
res
f
.
The radius rCD after unloading, Ru , is determined as
Ru
1 C1
C1 R02 R12
.
(12)
R0 2 R12 ln R1 R0
3 Material model
Let us consider the bending of orthotropic rolled sheet. The principal axes of or-
thotropy coincide with the coordinate lines of the Cartesian coordinate system: x, y
and z are thickness, transverse and rolling directions, respectively, i.e. in the case
under consideration the bending line coincides with rolling direction. Therefore, the
yield function proposed in [14] is:
23 xx yy 12 yy zz 31 zz xx
2 2 2
(13)
5 2 5 5 2
4 23 xy 12 2yz 31 zx 2 2
eq .
2 2 2
Here eq is the equivalent stress, ij are the components of the stress tensor in the
Cartesian system of coordinates. The generalized anisotropy factors ij are defined
as
15 A 1 1
ij 1 i j k , (14)
3 2 A 5
where A is the anisotropy factor of a crystal lattice, i are the parameters of the
crystallographic texture. The anisotropy factor is defined through the compliance
, S1122
constants of crystal lattice S1111
and S2323
S S
A 1111 1122 . (15)
2S2323
For a certain crystallographic orientation hkl uvw the parameters of the crystallo-
graphic texture are defined as
h2 k 2 ki2li2 li2 hi2
i i i , (16)
hi ki li
2 2 2 2
where hi , ki , li are Miller’s indices, which determine the i-th direction in the crystal
with respect to the principal axes of anisotropy.
Considering the tension along the principal axes, let us determine the tensile yield
stresses from Eq. (13):
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12 23
X Y 12 31 Z . (17)
23 31 23 31
Using Eq. (17), the last equation of (8) can be transformed into
Z 12 31
p , (18)
G 1223 2331 3112
where Z is the tensile yield stress in the rolling direction.
Thus, the material model considers the crystal lattice constants ( A ), i.e. chemical
composition of alloy, and the crystallographic texture ( i ), i.e. its thermo-mechanical
treatment. Using this model, it is possible to analyze the influence of the ideal crystal-
lographic orientations over the springback in bending.
4 Numerical results
Let us consider the copper sheet for which components of compliance tensor are
S1111
15.0 TPa-1; S1122
6.30 TPa-1 and S 2323 3.33 TPa-1 [15], i.e. A 3.203
(Eq. (15)); the tensile yield stress in the rolling direction Z is 340 MPa; the shear
modulus of elasticity G is 44 GPa. As an example, let us take the following ideal crys-
tallographic orientations: copper {112}<111>, brass {110}<112> (rolling compo-
nents) and Goss {110}<001> (recrystallization component). Also, as a comparison let
us consider the isotropic case, for which A 1 or i 1 5 [16], i.e. ij 1 .
The parameters of crystallographic texture i , generalized anisotropy factors factors
ij and parameter p calculated using Eq. (16), (14) and (18) for the stated compo-
nents are listed in Table 1.
Table 1. The parameters of crystallographic texture and the generalized anisotropy factors
of the single ideal components
Component The parameters of crys- The generalized anisot- p 103
tallographic texture ropy factors
Name Orientation 1 2 3 12 23 31
Copper {112}<111> 0.333 0.250 0.250 0.533 1.116 0.533 6.785
Brass {110}<112> 0.250 0.333 0.250 0.533 0.533 1.116 10.496
Goss {110}<001> 0.0 0.250 0.250 1.703 -0.054 1.703 15.969
Isotro- - 0.200 0.200 0.200 1.0 1.0 1.0 8.923
py
The dependence of the bending moment on the radius of the concave surface at the
beginning of the process is shown in Fig. 2. It is seen from this diagram that the effect
of anisotropy is revealed at the very beginning of the process only. Note that the re-
quired bending moment for some ideal crystallographic orientations can be bigger
({112}<111>) or smaller ({110}<112>, {110}<001>) in comparison with the iso-
tropic material.
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Fig. 2. Effect of ideal crystallographic orientations on the bending moment
The dependence of the springback Ru R0 on value of radius rCD is depicted in
Fig. 3. The figure shows that the springback decreases as the deformation at the end
of loading increases. All other things being equal the orientations {110}<112> and
{110}<001> cause bigger springback than the isotropic material and orientation
{112}<111>, for which the springback is minimal.
Fig. 3. Dependence of the springback on the radius at the end of loading
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The influence of the ideal crystallographic orientations upon the thickness distribution
of the residual stresses is illustrated in Fig. 4. In particular, the radial stress is shown
in Fig. 4a and the circumferential stress in Fig. 4b at rCD H 60 . The figures speci-
fied reveal that the effect of anisotropy is most significant at the central part of the
specimen. The distribution of the residual stresses outside the central part of the spec-
imen is almost independent from the crystallographic orientation except for
{110}<001>.
a) b)
Fig. 4. Through thickness distribution of the radial (a) and circumferential (b) residual stresses
at rCD H 60 at the end of loading
5 Conclusions
The plane strain pure bending of materials textured has been investigated. It has been
stated that depending on the ideal crystallographic orientations the springback and the
distribution of the residual stresses change grossly. Note that the springback of the
textured metal sheet can be bigger ({110}<112>, {110}<001>) or smaller
({112}<111>) in comparison with the isotropic material.
Acknowledgements
The reported study was funded by RFBR according to the research project №16-38-
00495.
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