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steel laminate sheets
ELSEVIER
Journal of Materials Processing Technology 52 (1995) 319 337
Journal of Materials Processing Technology
Finite-element simulation of the bending process of steel/polymer/steel laminate sheets You-Min Huang*, Daw-Kwei Leu Department of Mechanical Engineering, National Taiwan Institute of Technology, 43 Keelung Rd., Sec. 4, Taipei, Taiwan, ROC
Received 11 January 1994
Industrial Summary A steel/polymer/steel three-layer laminate has been developed for weight reduction and vibration-damping of automobile parts and panels. The bending process of the laminate sheets is simulated by an incremental elasto-plastic finite-element method under the plane-strain condition with care taken to describe the boundary conditions of penetration, separation and alternation of the sliding-stricking state of friction. The simulation demonstrates clearly the processes of the generation of a folding defect in the laminate sheet at the bent flange and a shear defect between the two skin steel layers along the entire length of the laminate sheet. The calculated sheet geometries and the punch load punch stroke relationship agree satisfactorily with the corresponding experimental results.
1. Introduction A steel/polymer/steel three-layer laminate is a new composite sheet that is used in several automobile parts and panels. Two types of steel/polymer/steel laminate with lightness and vibration-damping characteristics have been applied in the automobile industry, the structures of the laminate sheets being illustrated in Fig. 1. Despite reduction of weight, the light-weight laminate steel sheet can maintain a bending rigidity that is almost equal to that of a simple steel sheet of the same total thickness. The vibration-damping steel sheet has an excellent vibration-damping property. However, the steel/polymer/steel laminate shows poor formability, several geometrical defects occurring in forming processes, the defects being observed typically in the bending process. One example of the defects is the folding of the sheet at the bent flange, which does not arise in the bending of a simple steel sheet. Another example is a shear that occurs occasionally between two skin steel layers along the entire length of the sheet, even at the stage of non-severe deformation.
*Corresponding author. 0924-0136/95/$9.50 (t~ 1995 Elsevier Science S.A. All rights reserved SSDI 0 9 2 4 - 0 1 3 6 ( 9 4 ) 0 1 6 1 7 - A
320
E-M. Huang, D.-K. Leu /'Journal of Materials Processing Technology 52 (1995) 319 337
// /
(G)
(~'~ ')
~//
~,
,// / ~ V ' - ~ - - T J >'3 T
steu ~ki~
x
~
Steel skin
,i
STRUCTURE
I
\\ STRUCTUR~
,,
! [
]
\ Light weight Iominate /
I'--..R2
!
J
r
/ ~
/"
1,/2w. ~
0![ J
I
~%= (i or ~
~zrt~
mm
Fig. 1 Structure of two types of steel/polymer/steel laminate sheets and the dimensions of the tools and the sheet.
In the bending process, a large shear deformation occurs in the core, as the polymer core is weak compared with the metallic faces. In experimental work, Hayashi et al. [1] reported that the press formability of a laminated sheet is inferior to that of cold-rolled steel because of the extremely small deformation-resistance of the core film. Yoshida [-2] reported that folding was suppressed by the modification of the thickness ratio and the yield strength ratio of the skin steel. The deep drawability, stretch formability, stretch-flange formability, conical-shell formability and V-press bending behavior of the sheets were tested. Shinozaki et al. [3] described the characteristics of a steel plastic steel laminate, which is expected to contribute to the weight-saving of automobiles. Yutori [-4] reported that folding was suppressed by application of a wide-trenched die, however, the appropriate combination of material and die had not been found. Hirose et al. [5] concluded that decreased shear deformation of the core films suppresses press failures of a laminated sheet. Itoh et al. [6] investigated the elasto-plastic bending relevant to the press formability of sandwich plates. Makinouchi et al. [7] proposed an elasto-plastic finite-element computer program, based on the updated Lagrangian formulation, to simulate the entire bending process of steel-plastic laminate sheets without friction and to predict the folding defect. Yoshida [8] presented a new analytical solution for the elasto-plastic bending of metal/polymer/metal sandwich plates that takes into account the transverse shear of a polymer core. The purpose of this work is to provide an elasto-plastic finite-element computer program, based on the updated Lagrangian formulation, to simulate the entire bending process as plane-strain deformation and to predict precisely the geometrical
Y.-M. Huang, D.-K. Leu /Journal of Materials Processing Technology 52 (1995) 319-337 321 Table 1 The four different laminate sheets for experiment and simulation Type of laminate sheet [7]
Thickness of e a c h layer (mm)
Polymerin intermediate layer
tjtc/t~ Light-weight laminate steel s h e e t
Nylon-core
0.2/0.4/0.2
Nylon-6
Polypropylene-core 0.2/0.4/0.2
Polypropylene
Vibration-clamping steel sheet
For high temperature use
0.4/0.07/0.4
Acrylic acid copolymer
For normal temperature use
0.6/0.04/0.6
Polyester
defects of folding and shear. The exact boundary conditions, including the conditions of penetration, separation, alternation of the sliding sticking state of friction and corner contact, are formulated to treat the highly non-linear sheet tool contact problem in an incremental way. An extended r-minimum method is adopted to limit the size of each incremental step, in order to maintain the boundary conditions constant within each time step. The U-bending processes of four laminate sheets cited in Table 1 are simulated and the results are compared with the corresponding experimental results [7].
2. Description of the basic theory 2.1. Material model In order to model the deformation process of the steel/polymer/steel laminate sheet, the constitutive equation of the polymers used as the core material and the adhesive state of the steel-plastic interface are considered. The J2-flow constitutive equation *
zij -- 1 E+ v
I
6ik(~j I +
v ~
~ij(~k I - -
3~(E/(1 + v))a'iitr'kt q~ 262(3 H' + El(1 + v))] kl
(1)
is employed [-7] to model the elasto-plastic behaviour of both the polymers and the steel. In this equation, ~ij is the J a u m a n n rate of Kirchhoff stress, aij is the Cauchy stress, ~r~j is the deviatoric part of cr~, g~i is the rate of deformation which is the symmetric part of the velocity gradient L~j ( = #v~/OXj), X~ is the spatial fixed Cartesian coordinate, H' is the strain-hardening rate, E is Young's modulus, v is Poisson's ratio, and ~ assumes the value unity for the plastic state and zero for the elastic state or for unloading. The Jz-t]ow law is simple and all the material constants can be determined from only the uniaxial tension test.
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Table 2 M a t e r i a l c o n s t a n t s and characteristic r e l a t i o n s h i p s Material [7]
Yield Stress ( k g / m m 2)
Young's Modulus E ( k g / m m 2)
Poisson's ratio v
Stress plastic strain relationship ( k g / m m 2)
1. 2. 3. 4.
(ry = 26.4
Steel Nylon-6 Polypropylene Acrylic acid copolymer 5. Polyester
5.8 2.4 1.5
17 160 280 85 20
0.3 0.33 0.33 0.33
a = 55.9(0.02 + ~;p)0.193
a,cry = cry cr~ =
0.3
10
0.33
a - 0.12(0.5 + cp) ~2 + 0.25
cr -- 13.5(0. l + ~p)1.6 __ 5.46 a -- 3.0(0.1 + ~:p)2 ° + 2.37 a = 0.58(0.6 + ~:p)i.3 ._ 1.2
(1 k g / m m 2 = 9.81 N/ram2).
The equivalent stress-equivalent plastic strain relationship of polymers is represented by a modified n-power law of the form [7] 6=c(c o+%)"+b
(2)
in order to model the hardening behaviour of the polymers. The constants in Eq. (2) are determined from uniaxial tension test of both the sheet polymers used as the core material and the steel sheet used for the skin layer, and are given in Table 2 along with the elastic constants. Between the steel and the polymer, complete adhesion such that no peeling and no sliding exist at the interface is assumed in the calculation. The two finite-element meshes shown in Fig. 1 are used to model the structure of the vibration-damping steel sheets and the light-weight laminate steel sheets.
2.2. Saffhess equation Before considering contact problems, the process of discretization of the virtual work equation is discussed briefly. Adopting the updated Lagrangian formulation (ULF) in the framework of application of the incremental deformation for the metal forming process (bulk forming and sheet forming) is the most practical approach to describe the incremental characteristics of the plastic flow rule. The current configuration in the U L F at each deformation stage is used as the reference state to evaluate the deformation in a small time interval At such that the first-order theory is consistent with the accuracy requirement. The virtual work rate equation of the updated Lagrangian equation [9] is written as
f~ (~ij - 2aiki:kj)bi:ij dV + ft ajkLik(~LijdV =
~ [)&~dS,
(3)
where vi is the velocity; i/is the rate of the nominal traction; V and St- are the material volume and the surface on which the traction is prescribed.
E-M. Huang, D.-K. Leu/Journal of Materials Processing Technology 52 (1995) 319-337
323
It is assumed that the distribution of the velocity {v} in a discretized element is
{,,}
(4)
= eN]{d},
where [ N ] is the shape function matrix and {d} denotes the nodal velocity. The rate of deformation and the velocity gradient are written as {~: = [B]{d},
(5)
{L} -- EE]{d}.
(6)
[B] and [ E ] represent the strain rate-velocity matrix and the velocity gradient velocity matrix, respectively. Substituting Eqs. (5) and (6) into Eq. (3), the elemental stiffness matrix is obtained. As the principle of virtual work rate equation and the constitutive relationship are linear equations of rates, they can be replaced by increments defined with respect to any monotonously increasing measure, such as the tool-displacement increment. Following the standard procedure of finite elements to form the whole global stiffness matrix, EK]{Au} = { A F } ,
(7)
in which ?
In these equations, [ K ] is the global tangent stiffness matrix, [Dcv] is the elemental elasto-plastic constitutive matrix, Au denotes the nodal displacement increment, and IAF} denotes the prescribed nodal force increment. [Q] and [G] are defined as stress-correction matrices due to the current stress states at any stage of deformation.
3. Numerical analysis 3.1. Formulation o[ contact problems On some occasions the tool has a corner on which the sheet slides in forming, the simple but basic contact model shown in Fig. 2 is considered, in which line element IJ of the sheet slides on corner K of the tool. Here, for simplicity, a linear or bilinear element that has straight sides is used, so that the segment IJ is straight. At the contact interface the modified Coulomb friction law proposed by Oden and Pries [10] and Germain et al. [11] is assumed, two contact friction states, sticking and sliding, and discontinuous change of sliding direction, which latter can be treated well according to this friction law. Global coordinates (x,y) and local coordinates (I,n) are used to
324
E-M. Huang, D.-K. Leui"Journal o/Materials Processing Technolo~' 52 (1995) 319 337
/
/
J~'
at
time
t
I / /
at time t + d t Corner-contact
model
Fig. 2. The die c o r n e r - c o n t a c t model.
describe the nodal force, the displacement and the element stress, strain, etc. The/-axis is consistent with the longitudinal direction of the sheet, whilst the n-axis is perpendicular to the/-axis. 3.1.1. Friction condition
Special care is needed to treat the friction condition. The nodal force acting on the contact node of the sheet as it slides along the curved surface of the tool can be resolved into tangential and normal components [i andJj,: F =f,n
(8)
+ftl,
where n is the outward vector normal to the tool surface. The tangential force, the direction of which coincides with the sliding direction, is obtained by the modified C o u l o m b friction law .ll
•
rcl
(9)
+_+14[.O(Au I )
with the function ~b(Aul¢1) defined as /
A ircl
qS(Aufel) = tanh ( V ~
\
\
3]'
R/_/
(10)
in which VCRI is the limit displacement increment for quasi-sticking and auj - ,e~ is the sliding displacement increment relative to the tool movement: Au~ d = Aut - U sin O.
(11)
E-M. Huang, D.-K. Leu /Journal of Materials Processing Technology 52 (1995) 319-337
325
In the l - n local c o o r d i n a t e system, in which 0 is the angle between the/-axis and the horizontal axis, Aut is the tangential displacement increment of the contact node and U is the tool displacement. T h e increment of F is written as
AF = Af, n + f, A n + Af~t + f~At,
(12)
in which Au~el
An--
+
!,
(13)
n,
(14)
P Au~ el A! = TP
F
/ Au~°' \
f.a(Au~ e')
All=- _+/~{Af, t a n h ~ v ~ ) - ~
----7-
(VCR,/3tcosh2 I_
]
A rel ----'
(15)
tvL~J/J/_l
where p denotes the radius of the tool and _+ depend on the curvature of the tool. T h e a b o v e equations obviously include a correction term, which should be incorporated into the relative c o m p o n e n t of the global stiffness matrix with respect to the contact node by
f ( i - 1)
"" • .-
+J"
k,1 _
P
cIi ') k 2 1 -~J' p
"'"
k12
Aul
At7,
=
+f"('
P
+ Af, (~- ~)
_Tfl(i- X)U sin O k22
P
+
,
(16)
A f, (i)
in which Aft, is a prescribed displacement increment, equal to U cos 0.
3.1,2. Die-corner contact problems T h e distance ratio,/3, is defined as KJ/IJ. Assuming here that a tool with a corner is fixed in space, the relationships for nodes I, J and K, as shown in Fig. 2, are X K = X J -~- / 3 ( X 1 - - X J ) .
(17)
In the l n direction, the relationships for displacement increment are
(1
/3)Au~+/3Ad, = Au~,
(1 - / 3 ) A u J + / 3 A u ~, = 0,
(18) (19)
The following relationships between nodal n o r m a l forcesf, l , f J and f, K are assumed:
f.~ =L' +/L
(20)
f~, = [3.f~, fJ = (1 -- fl)f~.
(21)
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E-M. Huang, D.-K. Leu /'Journal of Materials Processing Technology 52 (1995) 319 337
F o r the incremental form, An = (Au'./(1 - fi)d)l and AI = incremental form of A P is written as
AFI=
[
a 1
f;
XI--
'
(1 -- ,q)d
z~/t / ] #it -t-
J
A f; I
.[tl
r
+ (1-/7)~
(ku~./'(1
i
fl)d)n; the
1
(22}
Au,, l,
where d is the length of segment I J, Af~r = flA.[',,K, and q
AJ, :
tel I) ff~'1A ( A lASt-,
rel'
]
- i , [ A f l ~ t a n h ( A.u]:'i~ + L \VLKI/3J (VCRI/3)cosh2(AuIeI'/VCRI/'3)J "
(23)
As above, the increment of form of A F j is
AF 1=
J}'Au,,jn+ ,1 A/,,j ~-/~
where A f d = (1
A.I,
[ A.t), I~ ,1 rid Au,,]l,
(24)
f l ) k f , K, and
t' AI~,Jtanh \ V C R T ] 3 } + ( V C R I / 3 ) c o s h 2 ( A u S / ( V C R 1 / 3 ) ) J "
(25)
Modification of the stiffness equation (16) according to the additional relationships, Eqs. (22) and (24), yields the matrix equation
..
k J1
kll2 +'[~(i
/~d
..
k J1
k{2
(J(i ]!
11 1I (i(i
k~lx k~ k~x
k~2 -~
11
"
{1 -/~)d ,/H 1, (1 -
fl)d
k,1 + t~
11 P
k21 ~-
,c{i .ll
P
k12
1)
/{22
Y.-M. Huang, D.-IC Leu /Journal of Materials Processing Technology 52 (1995) 319 337 327 ...
/~if/J0
1)
.o,
au~ au~ au', au'l
(1
-
f
A fit(i- 1) •
Aut A (tn •
fl)Af~ (')
OAr
]~ J n A f / ( i - 1)
[
(26)
K(i)
q_f~niD 1) U s i n ,,-(i
A f Ot ) 7~-- J !
~)
P
Usin0
Care must be taken, as the stiffness matrix in Eq. (26) is not symmetric. In the solution scheme, the contact nodal friction force increment components are evaluated iteratively; that is, Af~ is given at each iteration by the value obtained in the former iteration step.
3.2. Numerical analysis The analysis of the sheet-bending process is based on consideration of the planestrain condition. The contact processes of U-bending are simulated to demonstrate the applicability of the method presented here. Because of the symmetry of the plate, only the right-half portion of the tools and workpiece are modelled. An automatic mesh program was used to generate the finite-element mesh grid. Fig. 1 shows the profile of the die, the punch head, the initial shape and the finite-element mesh. A fine mesh is generated between the punch and the die to increase the accuracy of the simulation results. Teflon film lubricant is used in experimental work [7]. To realize satisfactory lubrication between the tool and the sheet, a friction coefficient/~ = 0.02 is assumed in the calculation.
3.3. Boundary condition Fig. 3 shows the sheet deformation geometry at a certain stage of the U-bending simulation. The boundary condition includes four portions: (a) The boundary on the AB section A.~4=0,
Af.¢0
and
Au.=A~i.,
where Af~ is the nodal tangential friction force increment; and zXf, is the normal force increment. As the material-tool contact area is assumed to involve friction, Af~ and A.f, are not equal to zero. Au,, which denotes the nodal displacement increment in the
328
E-M. Huang, D.-K. Leu/'Journal of Materials Processing Technology 52 (1995) 319 337
c
9,
IY
x
C1 C2 U=l .80mm Fig. 3. The boundary conditions for the deformed sheet geometry.
normal direction of the profile of the tools, is determined from the prescribed displacement increment of the punch Ag,. (b) The boundary on the BC, C1A and C2D (except corner K) section AFx=0
and
AFv=0.
The above condition reflects that the nodes on this boundary are free. (c) The boundary on the C1 Ca section Aux=0
and
AG=0.
The position of C1 C2 is the center-line of the sheet. Thus the displacement increment of the node in the x-direction is set unconditionally to zero. (d) The contact boundary of corner K: As the die radius used in experimental work [7] is very small, the die corner-contact model is assumed in the calculation. The corner contact nodal normal force increment (normal to segment I J, shown in Fig. 2) A f K is distributed to nodes I and d in the ratio fl: a f; = fiaf,
a.fd
= (1 - f l ) A f ,
The above relationships, instead of the corner contact normal force increment, are combined into the stiffness equations. As the bending process proceeds, the boundary becomes altered. For this reason the normal f o r c e r , of the contact nodes along boundary section AB at each deformation stage is examined. If fn reaches zero, the nodes become free and the boundary condition is shifted from portion (a) to (b). Meanwhile, the free nodes along the BC, C~A and C2D sections of the sheet are also verified in the computation. |f the node comes into contact with the punch, the free-boundary condition is altered to the constraint condition (a); at the die-sheet contact interface the corner-segment contact mode is used, i.e., a corner K of the die is in contact with segment IJ of the sheet, as shown in Fig. 2. As the constant strain triangular element is used in this calculation, the segment lJ is invariably a straight line.
Y.-M. Huang, D.-K. Leu /Journal of Materials Processing Technology 52 (1995) 319 337
329
3.4. Treatment o f the elasto-plastic and contact problems
From the interpretation of the former boundary condition, the contact condition remains unaltered within one incremental deformation process. In order to satisfy this requirement the r-minimum method proposed by Yamada [12] is adopted and extended to treat the elasto-plastic and contact problems [13]. The increment of each loading step is controlled by the smallest value of the following six values. (1) Elasto-plastic state: When the stress of a Gaussian point is greater than the yield stress, the value of r l is computed according to Eq. (27) to ascertain the stress just as the yield surface is reached [12]. rl =
B + (B 2 + 4 A C ) 1/2 2A
(27)
and 1
A = : [ F ( A a y - Aaz) 2 + G(Aaz - Aax) 2 + H ( A 6 x - Acrr) 2 + 2LAr2y],
B = A - 1 ( f + a + H)[2ftAft + (Aft)z], C = l ( v + G + H)(6Zy -- ftz),
where Aax, Aay, A ~ , A~y are the stress component increments, 6y is the yield stress, 6 and A6 are the equivalent stress and the equivalent stress increment, respectively. F, G, H and L are the anisotropic parameters. (2) The maximum strain increment: The value of rz is obtained from the ratio of the defaulted maximum strain increment ~ to the principal strain increment Ac., i.e.
r2 = ~
(28)
in order to limit the incremental step to such a size that the first-order theory is valid within the step. (3) The maximum rotation increment: In sheet bending, the accuracy of the simulation is influenced significantly by the rotation increment Aw. Because of this consideration, the value of r3 is calculated from the defaulted maximum rotation increment fl to the rotation increment Aw, r3 = A~'
(29)
in order to limit the incremental step to such a size that the first-order theory is valid within the step. (4) Penetration condition: When forming proceeds, the free nodes of the sheet may penetrate the tools. The ratio r 4 is calculated such that the free nodes just contact the tools. For instance, a segment of the tool is described by an arc. Then, r4 is calculated to let the node, which has coordinate (x, y) at the (i - 1)th step and (x + A u x , y + Auy)
330
E-M. Huang, D.-K. Leu /Journal of Materials Processing Technology 52 (1995) 319 337
at the (i)th step, reach the tool surface [13]: - "6 + (c 2 - c~c7) 1'2
r4 =
,
C5
(30)
in which C 5 = C 2 q- C 23 , C6 ~
C l C 2 -~- C 3 C 4 ,
"7 = c ~ + , ' ]
p2
and
cl = Aut -- Auy,
c2 = Yo + A u t R S U M - y, ':3 = Aux, C,,
X
--
X o .
Au, is the tool displacement increment, R S U M is the sum of the r-minimum values from the first step to the (i - 1)th step, p is the radius of the tool arc, and ( x o , y o ) is the coordinate of the center of the arc. (5) Separation condition: As it is assumed that the sheet and tools in contact do not adhere to each other, the nodes that contact with the tools m a y be separated from the contact surface. If the normal forceJ~, > 0 at time t, and./~, = 0 at time t + At, the value of r5 is obtained from [13] - - .}(~r
r5-
AJ~,
(31)
for each contact node such that the normal c o m p o n e n t of nodal force becomes zero. (6) Sliding-sticking friction condition: The modified C o u l o m b friction law gives two alternative contact states, sliding or sticking. Such states are verified for each contacting node according to the following conditions: (1) Au~eJm'Au~ e~(i (a) If
IAu~e~lil I >
1)
)
0.
VCRI, then ji-Ct./~,, the node is in a sliding state.
(b) If IAui¢~"~l < VCRI, then .[j-lt[',(Au~eJ/VCR1), the node is in a quasi-sticking state. (2) Z~u~eI(i)'z~U~ el(i 1) ( 0. W h e n the direction of a sliding node becomes opposite to the direction of the previous incremental step, the contact node is declared as a sticking node at the next incremental step. Then, a value of ratio r 6 is developed which gives the alteration of the friction
Y.-M. Huang, D.-K. Leu /Journal of Materia& Processing Technology 52 (1995) 319 337
331
state from sliding to sticking as Tolf r6 - iAu~el¢i)l,
(32)
where Tolf ( = 0.0001) is a small tolerance and Au~ el(1) is a fictitious relative displacement increment. The constants of the maximum strain increment ~ and maximum rotation increment ~ are taken to be 0.002 and 0.5 °, respectively; these prove to be valid in the first-order theory. Furthermore, a small tolerance in the procedure of verification of the conditions for penetration and separation is allowed. 3.5. Unloading process
In the sheet-forming process, the phenomenon of sprint-back or spring-go is significant, so that unloading following sheet forming is considered. The unloading procedure is executed by assuming that the nodes on the center-line are fixed in the x-direction; all elements are reset to be elastic. The force of the nodes that contact the tools is reversed to become the prescribed force boundary condition on the sheet, i.e., AF = -- F.
(33)
Meanwhile, the check of the conditions of penetration, friction and separation is excluded in the simulation program.
4. Results and discussion
Fig. 4 shows deformed geometries and nodal velocity distributions of the lightweight laminate steel sheet with a polypropylene core at five different bending stages, the calculated geometries shown in Fig. 4(a) being compared with corresponding experimental results I-7]. At the same bent stage, both geometries agree satisfactorily. At the bent flange of the laminate sheet, a folding defect is observed clearly in both calculation and experiment. The folding starts at a punch stroke of U = 0.5 mm and increasing its degree as bending proceeds. Strong shear deformation is clearly produced in the polypropylene layer in the die cavity but no shear deformation exists outside of the die edge. The folding defect at the bent flange is caused by the sudden change of shear strain in the polypropylene layer along the sheet. In Fig. 4(b) nodal velocity distributions show the correct deformation procedures of bending process. Fig. 5 shows the shear AS between two skin steel layers to clarify the magnitude of the shear deformation in the polypropylene core. The distribution of AS indicates the characteristic nature of the bending process of the laminate sheet, the results of calculation agreeing very well with those of experiment. Fig. 6 shows the punch load-punch stroke relationship good agreement between calculation and experiment [7] being obtained for this relationship.
332
E-M. Huang, D.-K. Leu /'Journal (?[Materials Processing Technolog), 52 (1995) 319 337
(a)
(b)
U 0.50mm
S U=l,OOmm
spr
Final After unloading
velocity After unloading
Fig. 4. Showing: (a) deformed geometries: and (b) nodal velocity distributions; in the bending process of light-weight laminate steel sheet with a polypropylene core {W = 6 mm).
Fig. 7 shows deformed geometries and nodal velocity distributions of the light-weight laminate steel sheet with a nylon-6 core at five different bending stages, the calculated geometries agreeing very well with those of experiment [7]. Because of the strong characteristic property of the nylon-6 core material, little folding of the sheet at the bend flange is observed in both calculation and experiment. Figs. 8 and 9 show deformed geometries and nodal velocity distributions of the vibration-damping laminate steel sheet for high temperature and normal temperature use at five different bending stages, good agreement between calculation and experiment being obtained in the deformed geometries. In Fig. 9 shear between the two skin steel layers is observed clearly along the entire length of the sheet in the bending of the
E-M. Huang, D.-K. Leu/Journal of Materials Processing Technology 52 (1995) 319 337
333
0.50 oo0OOExperiment _ _ Calculation
data data
at at
U:2.Smm U=25mm
0.40
0
E 0.30 E 0O %_
\"
o(
0 0
~o 0.20
<1 0.10
00
0.00
I
1
2.50
1.00
5.00
Distance
7.50
from
center
10.0o
(mm)
Fig. 5. Distribution of shear between the two skin steel layers along the sheet length at the stage of U = 2.5 mm for the light-weight laminate steel sheet with a polypropylene core.
175
Calculation 150 ,.¢_.
,,,/
125
.
-0 100 0 O v"
75
O_
50
tz~
25
0
11]
O.Oq
l i l t 1 1 1 1 1 1 1
0.50
l.O0
I
1.50
I
I
II
I
2.00
I
t
I
I
It
2.50
t
11
3.00
Punch stroke ( m m ) Fig. 6. Punch load-punch stroke relationships in bending process of a light-weight laminate steel sheet with polypropylene core (W = 6 ram).
334
E-M. Huang, D.-K. Leu/Journal o/'Materials Processing Technology 52 (1995) 319 337
(b)
U=O,5Omrq
I
O=l , 00turn
I F
~3
Final s h a p e After unloading
~ s p r Nodal After
Fig. 7. As for Fig. 4, but
for
0r~m
velocity unloading
a nylon-6 core.
vibration-damping steel sheet for normal temperature use in which a soft core material, polyester, is used.
5. Conclusions An elasto-plastic finite-element computer program based on the updated Lagrangian formulation is developed to simulate the bending process of steel/polymer/steel laminate sheets.
E-M. Huang, D.-K. Leu /Journal of Materials Processing Technology 52 (1995) 319 337
(a)
335
(b)
U=O5.Omrq U=l.OOmm
/ U=.5lOmm
,x
s~
Final After u n l o a d i n g
~ J
Ospr
Nodal v e l o c i t y After unloading
Fig. 8. Showing: (a) deformed geometries; and (b) nodal velocity distributions; in the bending process of vibration-damping laminate steel sheet for high temperature use (W = 6 ram).
The treatment of the contact problem between the tool and the workpiece is discussed, and a formulation and algorithms are developed based on a corner-contact mode with friction. Each laminate presents a different geometrical defect, folding of the flange and shear between the two skin steel layers. The calculation demonstrates clearly that the folding of the sheet at the bent flange is caused by the concentrated shear deformation in the polymer layer, and the calculated results agree satisfactorily with those of experiment. The calculation predicts satisfactorily the shear between the two skin steel layers along the entire length of the sheet.
336
E-M. Huang, D.-K. Leu/'Journal o[Materials Processing Technolog)' 52 (1995) 319 337 I i
[
(a)
(b)
U
000mm
U = 1 , OOmm
U
]
30mm
I
U:
Deformed
geometry
1 .5
Nodal
0ram velocity
Fig. 9. As for Fig. 8, but for normal temperature use and W = 8 mm.
This work has provided an improved understanding of the sheet-bending process of steel/polymer/steel laminate sheets.
References [1] Y. Hayashi, K. Takatani and H. Nagaii, Press formability of the laminate steel sheet, Proc. Japanese Sprinq Con[~.lot the Technoloqy ~1' Plasticity, sponsored by the Japanese Society of Mechanical Engineers. Narashino. Japan, 19 2l May, 1983, p. 29 [in Japanese}.
Y.-M. Huang, D.-K. Leu/Journal o f Materials Processing Technology 52 (1995) 319 337
337
[2] M. Yoshida, Press formability of vibration-damping sheets, J. JSTP, 26(291) (1985) 394 (in Japanese). [3] M. Shinozaki, Y. Matsumoto and K. Tsunoyama, Formability of steel plastic laminates, J. JSTP, 26(291) (1985) 409 (in Japanese). [4] Y, Yutori, Press formability of the laminated steel sheet, Sokeizai, 27(11) (1986) p. 10 (in Japanese). [5] Y. Hirose, M. Kojima, Y. Hayashi and C. Sudo, Bending performance of steel plastic steel laminated sheet, Adv. Techn. Plasticity, 3 (1990) 1519. [6] K. ltoh and T. Sagawa, Simple analysis of double-bent in V-type bending of steel/resin/steel laminates vibration damping sheet, J. JSTP, 30(346) (1989) 1490 (in Japanese). [7] A. Makinouchi, H. Ogawa and K. Hashimoto, Finite element simulation of bending process of steel plastic laminate sheets, J. JSTP, 29(330) (1988) 755 (in Japanese). [8] F. Yoshida, Elastic plastic analysis for bending of metal/polymer/metal sandwich plate, Adv. Techn. Plasticity, 3(1990) 1513. [9] R. M. McMeeking and J. R. Rice, Finite element formulations for problems of large elastic plastic deformation, Int. J. Solids Struct., 11 (1975) 601 606. [10] J. T. Oden and E. B. Pries, Nonlocal and nonlinear friction law and variational principles for contact problems in elasticity, J. Appl. Mech., 50 (1983) 67 76. [11] Y. Germain, K. Chung and R. H. Wagoner, A rigid viscoplastic finite element program for sheet metal forming analysis, Int. J. Mech. Sci., 31(1) (1989) 1 24. [12] Y. Yamada, N. Yoshimura and T. Sakurai, Plastic stress strain matrix and its application for the solution of elastic-plastic problems by the finite element method, lnt. J. Mech. Sci., 10 (1968) 343 354. [13] Y. M. Huang, Y. H. Lu and A. Makinouchi, Elasto-plastic finite element analysis of V-shape sheet bend, J. Mats. Proc. Techn., 35 (1992) 129 152.
Journal of Materials Processing Technology 52 (1995) 319 337
Journal of Materials Processing Technology
Finite-element simulation of the bending process of steel/polymer/steel laminate sheets You-Min Huang*, Daw-Kwei Leu Department of Mechanical Engineering, National Taiwan Institute of Technology, 43 Keelung Rd., Sec. 4, Taipei, Taiwan, ROC
Received 11 January 1994
Industrial Summary A steel/polymer/steel three-layer laminate has been developed for weight reduction and vibration-damping of automobile parts and panels. The bending process of the laminate sheets is simulated by an incremental elasto-plastic finite-element method under the plane-strain condition with care taken to describe the boundary conditions of penetration, separation and alternation of the sliding-stricking state of friction. The simulation demonstrates clearly the processes of the generation of a folding defect in the laminate sheet at the bent flange and a shear defect between the two skin steel layers along the entire length of the laminate sheet. The calculated sheet geometries and the punch load punch stroke relationship agree satisfactorily with the corresponding experimental results.
1. Introduction A steel/polymer/steel three-layer laminate is a new composite sheet that is used in several automobile parts and panels. Two types of steel/polymer/steel laminate with lightness and vibration-damping characteristics have been applied in the automobile industry, the structures of the laminate sheets being illustrated in Fig. 1. Despite reduction of weight, the light-weight laminate steel sheet can maintain a bending rigidity that is almost equal to that of a simple steel sheet of the same total thickness. The vibration-damping steel sheet has an excellent vibration-damping property. However, the steel/polymer/steel laminate shows poor formability, several geometrical defects occurring in forming processes, the defects being observed typically in the bending process. One example of the defects is the folding of the sheet at the bent flange, which does not arise in the bending of a simple steel sheet. Another example is a shear that occurs occasionally between two skin steel layers along the entire length of the sheet, even at the stage of non-severe deformation.
*Corresponding author. 0924-0136/95/$9.50 (t~ 1995 Elsevier Science S.A. All rights reserved SSDI 0 9 2 4 - 0 1 3 6 ( 9 4 ) 0 1 6 1 7 - A
320
E-M. Huang, D.-K. Leu /'Journal of Materials Processing Technology 52 (1995) 319 337
// /
(G)
(~'~ ')
~//
~,
,// / ~ V ' - ~ - - T J >'3 T
steu ~ki~
x
~
Steel skin
,i
STRUCTURE
I
\\ STRUCTUR~
,,
! [
]
\ Light weight Iominate /
I'--..R2
!
J
r
/ ~
/"
1,/2w. ~
0![ J
I
~%= (i or ~
~zrt~
mm
Fig. 1 Structure of two types of steel/polymer/steel laminate sheets and the dimensions of the tools and the sheet.
In the bending process, a large shear deformation occurs in the core, as the polymer core is weak compared with the metallic faces. In experimental work, Hayashi et al. [1] reported that the press formability of a laminated sheet is inferior to that of cold-rolled steel because of the extremely small deformation-resistance of the core film. Yoshida [-2] reported that folding was suppressed by the modification of the thickness ratio and the yield strength ratio of the skin steel. The deep drawability, stretch formability, stretch-flange formability, conical-shell formability and V-press bending behavior of the sheets were tested. Shinozaki et al. [3] described the characteristics of a steel plastic steel laminate, which is expected to contribute to the weight-saving of automobiles. Yutori [-4] reported that folding was suppressed by application of a wide-trenched die, however, the appropriate combination of material and die had not been found. Hirose et al. [5] concluded that decreased shear deformation of the core films suppresses press failures of a laminated sheet. Itoh et al. [6] investigated the elasto-plastic bending relevant to the press formability of sandwich plates. Makinouchi et al. [7] proposed an elasto-plastic finite-element computer program, based on the updated Lagrangian formulation, to simulate the entire bending process of steel-plastic laminate sheets without friction and to predict the folding defect. Yoshida [8] presented a new analytical solution for the elasto-plastic bending of metal/polymer/metal sandwich plates that takes into account the transverse shear of a polymer core. The purpose of this work is to provide an elasto-plastic finite-element computer program, based on the updated Lagrangian formulation, to simulate the entire bending process as plane-strain deformation and to predict precisely the geometrical
Y.-M. Huang, D.-K. Leu /Journal of Materials Processing Technology 52 (1995) 319-337 321 Table 1 The four different laminate sheets for experiment and simulation Type of laminate sheet [7]
Thickness of e a c h layer (mm)
Polymerin intermediate layer
tjtc/t~ Light-weight laminate steel s h e e t
Nylon-core
0.2/0.4/0.2
Nylon-6
Polypropylene-core 0.2/0.4/0.2
Polypropylene
Vibration-clamping steel sheet
For high temperature use
0.4/0.07/0.4
Acrylic acid copolymer
For normal temperature use
0.6/0.04/0.6
Polyester
defects of folding and shear. The exact boundary conditions, including the conditions of penetration, separation, alternation of the sliding sticking state of friction and corner contact, are formulated to treat the highly non-linear sheet tool contact problem in an incremental way. An extended r-minimum method is adopted to limit the size of each incremental step, in order to maintain the boundary conditions constant within each time step. The U-bending processes of four laminate sheets cited in Table 1 are simulated and the results are compared with the corresponding experimental results [7].
2. Description of the basic theory 2.1. Material model In order to model the deformation process of the steel/polymer/steel laminate sheet, the constitutive equation of the polymers used as the core material and the adhesive state of the steel-plastic interface are considered. The J2-flow constitutive equation *
zij -- 1 E+ v
I
6ik(~j I +
v ~
~ij(~k I - -
3~(E/(1 + v))a'iitr'kt q~ 262(3 H' + El(1 + v))] kl
(1)
is employed [-7] to model the elasto-plastic behaviour of both the polymers and the steel. In this equation, ~ij is the J a u m a n n rate of Kirchhoff stress, aij is the Cauchy stress, ~r~j is the deviatoric part of cr~, g~i is the rate of deformation which is the symmetric part of the velocity gradient L~j ( = #v~/OXj), X~ is the spatial fixed Cartesian coordinate, H' is the strain-hardening rate, E is Young's modulus, v is Poisson's ratio, and ~ assumes the value unity for the plastic state and zero for the elastic state or for unloading. The Jz-t]ow law is simple and all the material constants can be determined from only the uniaxial tension test.
322
E-M. Huang, D.-K. Leu/Journal (if'Materials" Processing Technology 52 (1995) 319 337
Table 2 M a t e r i a l c o n s t a n t s and characteristic r e l a t i o n s h i p s Material [7]
Yield Stress ( k g / m m 2)
Young's Modulus E ( k g / m m 2)
Poisson's ratio v
Stress plastic strain relationship ( k g / m m 2)
1. 2. 3. 4.
(ry = 26.4
Steel Nylon-6 Polypropylene Acrylic acid copolymer 5. Polyester
5.8 2.4 1.5
17 160 280 85 20
0.3 0.33 0.33 0.33
a = 55.9(0.02 + ~;p)0.193
a,cry = cry cr~ =
0.3
10
0.33
a - 0.12(0.5 + cp) ~2 + 0.25
cr -- 13.5(0. l + ~p)1.6 __ 5.46 a -- 3.0(0.1 + ~:p)2 ° + 2.37 a = 0.58(0.6 + ~:p)i.3 ._ 1.2
(1 k g / m m 2 = 9.81 N/ram2).
The equivalent stress-equivalent plastic strain relationship of polymers is represented by a modified n-power law of the form [7] 6=c(c o+%)"+b
(2)
in order to model the hardening behaviour of the polymers. The constants in Eq. (2) are determined from uniaxial tension test of both the sheet polymers used as the core material and the steel sheet used for the skin layer, and are given in Table 2 along with the elastic constants. Between the steel and the polymer, complete adhesion such that no peeling and no sliding exist at the interface is assumed in the calculation. The two finite-element meshes shown in Fig. 1 are used to model the structure of the vibration-damping steel sheets and the light-weight laminate steel sheets.
2.2. Saffhess equation Before considering contact problems, the process of discretization of the virtual work equation is discussed briefly. Adopting the updated Lagrangian formulation (ULF) in the framework of application of the incremental deformation for the metal forming process (bulk forming and sheet forming) is the most practical approach to describe the incremental characteristics of the plastic flow rule. The current configuration in the U L F at each deformation stage is used as the reference state to evaluate the deformation in a small time interval At such that the first-order theory is consistent with the accuracy requirement. The virtual work rate equation of the updated Lagrangian equation [9] is written as
f~ (~ij - 2aiki:kj)bi:ij dV + ft ajkLik(~LijdV =
~ [)&~dS,
(3)
where vi is the velocity; i/is the rate of the nominal traction; V and St- are the material volume and the surface on which the traction is prescribed.
E-M. Huang, D.-K. Leu/Journal of Materials Processing Technology 52 (1995) 319-337
323
It is assumed that the distribution of the velocity {v} in a discretized element is
{,,}
(4)
= eN]{d},
where [ N ] is the shape function matrix and {d} denotes the nodal velocity. The rate of deformation and the velocity gradient are written as {~: = [B]{d},
(5)
{L} -- EE]{d}.
(6)
[B] and [ E ] represent the strain rate-velocity matrix and the velocity gradient velocity matrix, respectively. Substituting Eqs. (5) and (6) into Eq. (3), the elemental stiffness matrix is obtained. As the principle of virtual work rate equation and the constitutive relationship are linear equations of rates, they can be replaced by increments defined with respect to any monotonously increasing measure, such as the tool-displacement increment. Following the standard procedure of finite elements to form the whole global stiffness matrix, EK]{Au} = { A F } ,
(7)
in which ?
In these equations, [ K ] is the global tangent stiffness matrix, [Dcv] is the elemental elasto-plastic constitutive matrix, Au denotes the nodal displacement increment, and IAF} denotes the prescribed nodal force increment. [Q] and [G] are defined as stress-correction matrices due to the current stress states at any stage of deformation.
3. Numerical analysis 3.1. Formulation o[ contact problems On some occasions the tool has a corner on which the sheet slides in forming, the simple but basic contact model shown in Fig. 2 is considered, in which line element IJ of the sheet slides on corner K of the tool. Here, for simplicity, a linear or bilinear element that has straight sides is used, so that the segment IJ is straight. At the contact interface the modified Coulomb friction law proposed by Oden and Pries [10] and Germain et al. [11] is assumed, two contact friction states, sticking and sliding, and discontinuous change of sliding direction, which latter can be treated well according to this friction law. Global coordinates (x,y) and local coordinates (I,n) are used to
324
E-M. Huang, D.-K. Leui"Journal o/Materials Processing Technolo~' 52 (1995) 319 337
/
/
J~'
at
time
t
I / /
at time t + d t Corner-contact
model
Fig. 2. The die c o r n e r - c o n t a c t model.
describe the nodal force, the displacement and the element stress, strain, etc. The/-axis is consistent with the longitudinal direction of the sheet, whilst the n-axis is perpendicular to the/-axis. 3.1.1. Friction condition
Special care is needed to treat the friction condition. The nodal force acting on the contact node of the sheet as it slides along the curved surface of the tool can be resolved into tangential and normal components [i andJj,: F =f,n
(8)
+ftl,
where n is the outward vector normal to the tool surface. The tangential force, the direction of which coincides with the sliding direction, is obtained by the modified C o u l o m b friction law .ll
•
rcl
(9)
+_+14[.O(Au I )
with the function ~b(Aul¢1) defined as /
A ircl
qS(Aufel) = tanh ( V ~
\
\
3]'
R/_/
(10)
in which VCRI is the limit displacement increment for quasi-sticking and auj - ,e~ is the sliding displacement increment relative to the tool movement: Au~ d = Aut - U sin O.
(11)
E-M. Huang, D.-K. Leu /Journal of Materials Processing Technology 52 (1995) 319-337
325
In the l - n local c o o r d i n a t e system, in which 0 is the angle between the/-axis and the horizontal axis, Aut is the tangential displacement increment of the contact node and U is the tool displacement. T h e increment of F is written as
AF = Af, n + f, A n + Af~t + f~At,
(12)
in which Au~el
An--
+
!,
(13)
n,
(14)
P Au~ el A! = TP
F
/ Au~°' \
f.a(Au~ e')
All=- _+/~{Af, t a n h ~ v ~ ) - ~
----7-
(VCR,/3tcosh2 I_
]
A rel ----'
(15)
tvL~J/J/_l
where p denotes the radius of the tool and _+ depend on the curvature of the tool. T h e a b o v e equations obviously include a correction term, which should be incorporated into the relative c o m p o n e n t of the global stiffness matrix with respect to the contact node by
f ( i - 1)
"" • .-
+J"
k,1 _
P
cIi ') k 2 1 -~J' p
"'"
k12
Aul
At7,
=
+f"('
P
+ Af, (~- ~)
_Tfl(i- X)U sin O k22
P
+
,
(16)
A f, (i)
in which Aft, is a prescribed displacement increment, equal to U cos 0.
3.1,2. Die-corner contact problems T h e distance ratio,/3, is defined as KJ/IJ. Assuming here that a tool with a corner is fixed in space, the relationships for nodes I, J and K, as shown in Fig. 2, are X K = X J -~- / 3 ( X 1 - - X J ) .
(17)
In the l n direction, the relationships for displacement increment are
(1
/3)Au~+/3Ad, = Au~,
(1 - / 3 ) A u J + / 3 A u ~, = 0,
(18) (19)
The following relationships between nodal n o r m a l forcesf, l , f J and f, K are assumed:
f.~ =L' +/L
(20)
f~, = [3.f~, fJ = (1 -- fl)f~.
(21)
326
E-M. Huang, D.-K. Leu /'Journal of Materials Processing Technology 52 (1995) 319 337
F o r the incremental form, An = (Au'./(1 - fi)d)l and AI = incremental form of A P is written as
AFI=
[
a 1
f;
XI--
'
(1 -- ,q)d
z~/t / ] #it -t-
J
A f; I
.[tl
r
+ (1-/7)~
(ku~./'(1
i
fl)d)n; the
1
(22}
Au,, l,
where d is the length of segment I J, Af~r = flA.[',,K, and q
AJ, :
tel I) ff~'1A ( A lASt-,
rel'
]
- i , [ A f l ~ t a n h ( A.u]:'i~ + L \VLKI/3J (VCRI/3)cosh2(AuIeI'/VCRI/'3)J "
(23)
As above, the increment of form of A F j is
AF 1=
J}'Au,,jn+ ,1 A/,,j ~-/~
where A f d = (1
A.I,
[ A.t), I~ ,1 rid Au,,]l,
(24)
f l ) k f , K, and
t' AI~,Jtanh \ V C R T ] 3 } + ( V C R I / 3 ) c o s h 2 ( A u S / ( V C R 1 / 3 ) ) J "
(25)
Modification of the stiffness equation (16) according to the additional relationships, Eqs. (22) and (24), yields the matrix equation
..
k J1
kll2 +'[~(i
/~d
..
k J1
k{2
(J(i ]!
11 1I (i(i
k~lx k~ k~x
k~2 -~
11
"
{1 -/~)d ,/H 1, (1 -
fl)d
k,1 + t~
11 P
k21 ~-
,c{i .ll
P
k12
1)
/{22
Y.-M. Huang, D.-IC Leu /Journal of Materials Processing Technology 52 (1995) 319 337 327 ...
/~if/J0
1)
.o,
au~ au~ au', au'l
(1
-
f
A fit(i- 1) •
Aut A (tn •
fl)Af~ (')
OAr
]~ J n A f / ( i - 1)
[
(26)
K(i)
q_f~niD 1) U s i n ,,-(i
A f Ot ) 7~-- J !
~)
P
Usin0
Care must be taken, as the stiffness matrix in Eq. (26) is not symmetric. In the solution scheme, the contact nodal friction force increment components are evaluated iteratively; that is, Af~ is given at each iteration by the value obtained in the former iteration step.
3.2. Numerical analysis The analysis of the sheet-bending process is based on consideration of the planestrain condition. The contact processes of U-bending are simulated to demonstrate the applicability of the method presented here. Because of the symmetry of the plate, only the right-half portion of the tools and workpiece are modelled. An automatic mesh program was used to generate the finite-element mesh grid. Fig. 1 shows the profile of the die, the punch head, the initial shape and the finite-element mesh. A fine mesh is generated between the punch and the die to increase the accuracy of the simulation results. Teflon film lubricant is used in experimental work [7]. To realize satisfactory lubrication between the tool and the sheet, a friction coefficient/~ = 0.02 is assumed in the calculation.
3.3. Boundary condition Fig. 3 shows the sheet deformation geometry at a certain stage of the U-bending simulation. The boundary condition includes four portions: (a) The boundary on the AB section A.~4=0,
Af.¢0
and
Au.=A~i.,
where Af~ is the nodal tangential friction force increment; and zXf, is the normal force increment. As the material-tool contact area is assumed to involve friction, Af~ and A.f, are not equal to zero. Au,, which denotes the nodal displacement increment in the
328
E-M. Huang, D.-K. Leu/'Journal of Materials Processing Technology 52 (1995) 319 337
c
9,
IY
x
C1 C2 U=l .80mm Fig. 3. The boundary conditions for the deformed sheet geometry.
normal direction of the profile of the tools, is determined from the prescribed displacement increment of the punch Ag,. (b) The boundary on the BC, C1A and C2D (except corner K) section AFx=0
and
AFv=0.
The above condition reflects that the nodes on this boundary are free. (c) The boundary on the C1 Ca section Aux=0
and
AG=0.
The position of C1 C2 is the center-line of the sheet. Thus the displacement increment of the node in the x-direction is set unconditionally to zero. (d) The contact boundary of corner K: As the die radius used in experimental work [7] is very small, the die corner-contact model is assumed in the calculation. The corner contact nodal normal force increment (normal to segment I J, shown in Fig. 2) A f K is distributed to nodes I and d in the ratio fl: a f; = fiaf,
a.fd
= (1 - f l ) A f ,
The above relationships, instead of the corner contact normal force increment, are combined into the stiffness equations. As the bending process proceeds, the boundary becomes altered. For this reason the normal f o r c e r , of the contact nodes along boundary section AB at each deformation stage is examined. If fn reaches zero, the nodes become free and the boundary condition is shifted from portion (a) to (b). Meanwhile, the free nodes along the BC, C~A and C2D sections of the sheet are also verified in the computation. |f the node comes into contact with the punch, the free-boundary condition is altered to the constraint condition (a); at the die-sheet contact interface the corner-segment contact mode is used, i.e., a corner K of the die is in contact with segment IJ of the sheet, as shown in Fig. 2. As the constant strain triangular element is used in this calculation, the segment lJ is invariably a straight line.
Y.-M. Huang, D.-K. Leu /Journal of Materials Processing Technology 52 (1995) 319 337
329
3.4. Treatment o f the elasto-plastic and contact problems
From the interpretation of the former boundary condition, the contact condition remains unaltered within one incremental deformation process. In order to satisfy this requirement the r-minimum method proposed by Yamada [12] is adopted and extended to treat the elasto-plastic and contact problems [13]. The increment of each loading step is controlled by the smallest value of the following six values. (1) Elasto-plastic state: When the stress of a Gaussian point is greater than the yield stress, the value of r l is computed according to Eq. (27) to ascertain the stress just as the yield surface is reached [12]. rl =
B + (B 2 + 4 A C ) 1/2 2A
(27)
and 1
A = : [ F ( A a y - Aaz) 2 + G(Aaz - Aax) 2 + H ( A 6 x - Acrr) 2 + 2LAr2y],
B = A - 1 ( f + a + H)[2ftAft + (Aft)z], C = l ( v + G + H)(6Zy -- ftz),
where Aax, Aay, A ~ , A~y are the stress component increments, 6y is the yield stress, 6 and A6 are the equivalent stress and the equivalent stress increment, respectively. F, G, H and L are the anisotropic parameters. (2) The maximum strain increment: The value of rz is obtained from the ratio of the defaulted maximum strain increment ~ to the principal strain increment Ac., i.e.
r2 = ~
(28)
in order to limit the incremental step to such a size that the first-order theory is valid within the step. (3) The maximum rotation increment: In sheet bending, the accuracy of the simulation is influenced significantly by the rotation increment Aw. Because of this consideration, the value of r3 is calculated from the defaulted maximum rotation increment fl to the rotation increment Aw, r3 = A~'
(29)
in order to limit the incremental step to such a size that the first-order theory is valid within the step. (4) Penetration condition: When forming proceeds, the free nodes of the sheet may penetrate the tools. The ratio r 4 is calculated such that the free nodes just contact the tools. For instance, a segment of the tool is described by an arc. Then, r4 is calculated to let the node, which has coordinate (x, y) at the (i - 1)th step and (x + A u x , y + Auy)
330
E-M. Huang, D.-K. Leu /Journal of Materials Processing Technology 52 (1995) 319 337
at the (i)th step, reach the tool surface [13]: - "6 + (c 2 - c~c7) 1'2
r4 =
,
C5
(30)
in which C 5 = C 2 q- C 23 , C6 ~
C l C 2 -~- C 3 C 4 ,
"7 = c ~ + , ' ]
p2
and
cl = Aut -- Auy,
c2 = Yo + A u t R S U M - y, ':3 = Aux, C,,
X
--
X o .
Au, is the tool displacement increment, R S U M is the sum of the r-minimum values from the first step to the (i - 1)th step, p is the radius of the tool arc, and ( x o , y o ) is the coordinate of the center of the arc. (5) Separation condition: As it is assumed that the sheet and tools in contact do not adhere to each other, the nodes that contact with the tools m a y be separated from the contact surface. If the normal forceJ~, > 0 at time t, and./~, = 0 at time t + At, the value of r5 is obtained from [13] - - .}(~r
r5-
AJ~,
(31)
for each contact node such that the normal c o m p o n e n t of nodal force becomes zero. (6) Sliding-sticking friction condition: The modified C o u l o m b friction law gives two alternative contact states, sliding or sticking. Such states are verified for each contacting node according to the following conditions: (1) Au~eJm'Au~ e~(i (a) If
IAu~e~lil I >
1)
)
0.
VCRI, then ji-Ct./~,, the node is in a sliding state.
(b) If IAui¢~"~l < VCRI, then .[j-lt[',(Au~eJ/VCR1), the node is in a quasi-sticking state. (2) Z~u~eI(i)'z~U~ el(i 1) ( 0. W h e n the direction of a sliding node becomes opposite to the direction of the previous incremental step, the contact node is declared as a sticking node at the next incremental step. Then, a value of ratio r 6 is developed which gives the alteration of the friction
Y.-M. Huang, D.-K. Leu /Journal of Materia& Processing Technology 52 (1995) 319 337
331
state from sliding to sticking as Tolf r6 - iAu~el¢i)l,
(32)
where Tolf ( = 0.0001) is a small tolerance and Au~ el(1) is a fictitious relative displacement increment. The constants of the maximum strain increment ~ and maximum rotation increment ~ are taken to be 0.002 and 0.5 °, respectively; these prove to be valid in the first-order theory. Furthermore, a small tolerance in the procedure of verification of the conditions for penetration and separation is allowed. 3.5. Unloading process
In the sheet-forming process, the phenomenon of sprint-back or spring-go is significant, so that unloading following sheet forming is considered. The unloading procedure is executed by assuming that the nodes on the center-line are fixed in the x-direction; all elements are reset to be elastic. The force of the nodes that contact the tools is reversed to become the prescribed force boundary condition on the sheet, i.e., AF = -- F.
(33)
Meanwhile, the check of the conditions of penetration, friction and separation is excluded in the simulation program.
4. Results and discussion
Fig. 4 shows deformed geometries and nodal velocity distributions of the lightweight laminate steel sheet with a polypropylene core at five different bending stages, the calculated geometries shown in Fig. 4(a) being compared with corresponding experimental results I-7]. At the same bent stage, both geometries agree satisfactorily. At the bent flange of the laminate sheet, a folding defect is observed clearly in both calculation and experiment. The folding starts at a punch stroke of U = 0.5 mm and increasing its degree as bending proceeds. Strong shear deformation is clearly produced in the polypropylene layer in the die cavity but no shear deformation exists outside of the die edge. The folding defect at the bent flange is caused by the sudden change of shear strain in the polypropylene layer along the sheet. In Fig. 4(b) nodal velocity distributions show the correct deformation procedures of bending process. Fig. 5 shows the shear AS between two skin steel layers to clarify the magnitude of the shear deformation in the polypropylene core. The distribution of AS indicates the characteristic nature of the bending process of the laminate sheet, the results of calculation agreeing very well with those of experiment. Fig. 6 shows the punch load-punch stroke relationship good agreement between calculation and experiment [7] being obtained for this relationship.
332
E-M. Huang, D.-K. Leu /'Journal (?[Materials Processing Technolog), 52 (1995) 319 337
(a)
(b)
U 0.50mm
S U=l,OOmm
spr
Final After unloading
velocity After unloading
Fig. 4. Showing: (a) deformed geometries: and (b) nodal velocity distributions; in the bending process of light-weight laminate steel sheet with a polypropylene core {W = 6 mm).
Fig. 7 shows deformed geometries and nodal velocity distributions of the light-weight laminate steel sheet with a nylon-6 core at five different bending stages, the calculated geometries agreeing very well with those of experiment [7]. Because of the strong characteristic property of the nylon-6 core material, little folding of the sheet at the bend flange is observed in both calculation and experiment. Figs. 8 and 9 show deformed geometries and nodal velocity distributions of the vibration-damping laminate steel sheet for high temperature and normal temperature use at five different bending stages, good agreement between calculation and experiment being obtained in the deformed geometries. In Fig. 9 shear between the two skin steel layers is observed clearly along the entire length of the sheet in the bending of the
E-M. Huang, D.-K. Leu/Journal of Materials Processing Technology 52 (1995) 319 337
333
0.50 oo0OOExperiment _ _ Calculation
data data
at at
U:2.Smm U=25mm
0.40
0
E 0.30 E 0O %_
\"
o(
0 0
~o 0.20
<1 0.10
00
0.00
I
1
2.50
1.00
5.00
Distance
7.50
from
center
10.0o
(mm)
Fig. 5. Distribution of shear between the two skin steel layers along the sheet length at the stage of U = 2.5 mm for the light-weight laminate steel sheet with a polypropylene core.
175
Calculation 150 ,.¢_.
,,,/
125
.
-0 100 0 O v"
75
O_
50
tz~
25
0
11]
O.Oq
l i l t 1 1 1 1 1 1 1
0.50
l.O0
I
1.50
I
I
II
I
2.00
I
t
I
I
It
2.50
t
11
3.00
Punch stroke ( m m ) Fig. 6. Punch load-punch stroke relationships in bending process of a light-weight laminate steel sheet with polypropylene core (W = 6 ram).
334
E-M. Huang, D.-K. Leu/Journal o/'Materials Processing Technology 52 (1995) 319 337
(b)
U=O,5Omrq
I
O=l , 00turn
I F
~3
Final s h a p e After unloading
~ s p r Nodal After
Fig. 7. As for Fig. 4, but
for
0r~m
velocity unloading
a nylon-6 core.
vibration-damping steel sheet for normal temperature use in which a soft core material, polyester, is used.
5. Conclusions An elasto-plastic finite-element computer program based on the updated Lagrangian formulation is developed to simulate the bending process of steel/polymer/steel laminate sheets.
E-M. Huang, D.-K. Leu /Journal of Materials Processing Technology 52 (1995) 319 337
(a)
335
(b)
U=O5.Omrq U=l.OOmm
/ U=.5lOmm
,x
s~
Final After u n l o a d i n g
~ J
Ospr
Nodal v e l o c i t y After unloading
Fig. 8. Showing: (a) deformed geometries; and (b) nodal velocity distributions; in the bending process of vibration-damping laminate steel sheet for high temperature use (W = 6 ram).
The treatment of the contact problem between the tool and the workpiece is discussed, and a formulation and algorithms are developed based on a corner-contact mode with friction. Each laminate presents a different geometrical defect, folding of the flange and shear between the two skin steel layers. The calculation demonstrates clearly that the folding of the sheet at the bent flange is caused by the concentrated shear deformation in the polymer layer, and the calculated results agree satisfactorily with those of experiment. The calculation predicts satisfactorily the shear between the two skin steel layers along the entire length of the sheet.
336
E-M. Huang, D.-K. Leu/'Journal o[Materials Processing Technolog)' 52 (1995) 319 337 I i
[
(a)
(b)
U
000mm
U = 1 , OOmm
U
]
30mm
I
U:
Deformed
geometry
1 .5
Nodal
0ram velocity
Fig. 9. As for Fig. 8, but for normal temperature use and W = 8 mm.
This work has provided an improved understanding of the sheet-bending process of steel/polymer/steel laminate sheets.
References [1] Y. Hayashi, K. Takatani and H. Nagaii, Press formability of the laminate steel sheet, Proc. Japanese Sprinq Con[~.lot the Technoloqy ~1' Plasticity, sponsored by the Japanese Society of Mechanical Engineers. Narashino. Japan, 19 2l May, 1983, p. 29 [in Japanese}.
Y.-M. Huang, D.-K. Leu/Journal o f Materials Processing Technology 52 (1995) 319 337
337
[2] M. Yoshida, Press formability of vibration-damping sheets, J. JSTP, 26(291) (1985) 394 (in Japanese). [3] M. Shinozaki, Y. Matsumoto and K. Tsunoyama, Formability of steel plastic laminates, J. JSTP, 26(291) (1985) 409 (in Japanese). [4] Y, Yutori, Press formability of the laminated steel sheet, Sokeizai, 27(11) (1986) p. 10 (in Japanese). [5] Y. Hirose, M. Kojima, Y. Hayashi and C. Sudo, Bending performance of steel plastic steel laminated sheet, Adv. Techn. Plasticity, 3 (1990) 1519. [6] K. ltoh and T. Sagawa, Simple analysis of double-bent in V-type bending of steel/resin/steel laminates vibration damping sheet, J. JSTP, 30(346) (1989) 1490 (in Japanese). [7] A. Makinouchi, H. Ogawa and K. Hashimoto, Finite element simulation of bending process of steel plastic laminate sheets, J. JSTP, 29(330) (1988) 755 (in Japanese). [8] F. Yoshida, Elastic plastic analysis for bending of metal/polymer/metal sandwich plate, Adv. Techn. Plasticity, 3(1990) 1513. [9] R. M. McMeeking and J. R. Rice, Finite element formulations for problems of large elastic plastic deformation, Int. J. Solids Struct., 11 (1975) 601 606. [10] J. T. Oden and E. B. Pries, Nonlocal and nonlinear friction law and variational principles for contact problems in elasticity, J. Appl. Mech., 50 (1983) 67 76. [11] Y. Germain, K. Chung and R. H. Wagoner, A rigid viscoplastic finite element program for sheet metal forming analysis, Int. J. Mech. Sci., 31(1) (1989) 1 24. [12] Y. Yamada, N. Yoshimura and T. Sakurai, Plastic stress strain matrix and its application for the solution of elastic-plastic problems by the finite element method, lnt. J. Mech. Sci., 10 (1968) 343 354. [13] Y. M. Huang, Y. H. Lu and A. Makinouchi, Elasto-plastic finite element analysis of V-shape sheet bend, J. Mats. Proc. Techn., 35 (1992) 129 152.