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A finite-element analysis of sheet metal forming processes
Joura~of
Materials Processing Technology Journal of Materials Processing Technology 54 (1995) 70-75
ELSEVIER
A finite-element analysis of sheet metal forming processes Tze-Chi Hsu*, Chan-Hung Chu Department of Mechanical Engineering, Yuan-He Institute of Technology, Far East Rd., Tao Yuan, Taiwan 32026, ROC
Received 1 April
Industrial summary
A computational method based on the membrane theory for the analysis of axisymmetricsheet metal forming processes such as punch stretching, deep drawing and hydroforming is presented. The elastic-plastic finite element approach is based on the flow rule associated with Hill's quadratic yield criterion for anisotropic material. The total Lagrangian formulation and virtual work theory are used to derive the stiffnessequations and the relationship between displacementand strain. Some examples on the stretching,drawing and hydroforming of metal sheets are considered,and the computed results are compared with experimental data and with the results from existing numerical solutions.
1. Introduction
One of the first numerical analyses of the sheet metal forming process was done by Woo [1], who considered the material as isotropic and used a finite-difference method to solve the case of axisymmetric punch stretching and drawing. Although his results show good agreement with experimental data in the flange section, a significant discrepancy was observed in the thickness-strain distribution in the tooling/workpiece contact region, this phenomena being attributed to the changing friction conditions. Kobayashi [-2] and his co-workers have worked on several sheet forming problems including drawing, bulging, stretch forming and bending, solving the problems using a finite-element method based on incremental strain theory. In their work, the material was assumed as rigid-plastic (elastic deformation was neglected) and anisotropy and work-hardening characteristics were included. An elastic-plastic F.E.M. formulation of stretch forming for a punch and a die of arbitrary shape was introduced by Wang and Budiansky [3]. The material was governed by the Ramberg-Osgood relationship. Complete boundary conditions covering both the sliding and the sticking condition were provided. Comparisons were made by checking the case of hemispherical punch stretching with rigid-plastic results [-4] where it was noted that the strain distributions were essentially the same until the onset of material unloading in the punch contact region. The explicit form of the finite-element
*Correspondingauthor. 0924-0136/95/$09.50 © 1995Elsevier ScienceS.A. All rights reserved SSDI 0924-0136(95)01922-2
stiffness equations for the hemispherical stretching case was provided by Wang and Wenner [5]. An elastic-viscoplastic formulation which included strain-rate effects was given also by Wang and Wenner [6] and showed that rate sensitivity yields some benefits in punch stretching. Another numerical simulation based on membrane theory was derived by Tang [-7], who used a tensor formulation and applied finite inelastic strain theory to solve a flanged-hole forming problem with four different punch shapes. The results revealed that the strain path during the forming process is not affected by the punch shape but that the maximum punch load depends on the punch shape. All the numerical analysis mentioned above were obtained using simple membrane theory which does not include the variation of stresses through the sheet thickness. The assumption is appropriate in many sheet metal forming operations because the sheet thickness is usually small in comparison with the radius of either the punch or the die. However, in order to simulate bending phenomena such as the early deformation in stretch forming, spring-back etc., inclusion of the bending stress is necessary. Wifi [8] employed an incremental variation method to analyze axisymmetric elastic plastic solids at large strains. The method was applied to the problems of stretch forming and complete deep-drawing of a circular blank using a hemispherical punch. The contact problem in different zones was formulated and tackled by the finite-element method. Although the contact problem at the blank hoder, die, die profile and punch head were included, only a frictionless condition at the tooling/workpiece interface was assumed.
T.-C. Hsu, C.-H. Chu / Journal of Materials Processing Technology 54 (1995) 70-75
Triantafyllidis and Samanta [9] derived a total Lagrangian F.E.M. formulation for large displacements and rotations and large strain deformations of the shell of revolution under axisymmetric loading. The equilibrium equations were derived from the principal of virtual work and a frictionless condition was assumed. Another bending F.E.M. based on the total Lagrangian formulation was derived by Wang and Tang [10], a conventional displacement method being used to construct the principal of virtual work. Finite-element algorithms for axisymmetric and plane-strain cases were presented. The difference between the strain distributions for the membrane and bending results is quite small for axisymmetric punch stretching and cup drawing with typical tool radii. In the present work an analysis of a sheet metal operation involving punch stretching, drawing and hydroforming is developed using the finite-element method. To analyze these processes mathematically, the non-linearities caused by the elastic-plastic material property, and the large strains and displacements, must be considered. The Lagrangian description of motion is applied to the finite-deformation analysis of the thin sheet structure. Only membrane strains are considered. The plane-stress assumption (the stress component normal to the sheet is negligible) is adopted. Since the coordinates used here are the principal directions, only two stress- and strain-components are used in the basic equations. The sheet material is assumed to satisfy a rate insensitive, Mises-type flow rule which takes into account work hardening and anisotropy. At the tool-sheet interface, the friction stress is calculated from the coulomb friction law whenever relative sliding occurs.
o/
Ini
e,< = ln(2,,),
(3)
where ~ = 1, 2. The strain rates are defined as
,~==
-/~
(4)
where • denots the differential with respect to a time-like variable such as punch travel.
2.2. Constitutive equations The model treats plastic flow by a modified von Mises flow rule which satisfy Hill's rate-insensitive normal anisotropic relationships [7]. The strain rates consist of an elastic part and a plastic part, given by
where
~+u
- -
{/~(P'}
~,d° ~
1[1
.~-'/~(lp)~
,.~ee,
(5)
--v][lz~ ' Et
1
(6)
1 -I-- R
"el
1VR and
2R
~1/2 •
(8)
(1)
and
22 =
X
where subscript ¢ represents partial differential with respect to the initial radial position, ~. The principal logarithmic strains el and e2, in the radial and circumferential directions are respectively,
2.1. Displacement and strain equations
~lZ
u
Fig. 1. Initial and current positions of material in a shell.
{e} = {e~'~} + {e'~},
& = [(1 + u¢) 2 + w~]
Cz
%
2. Finite element formulation
A schematic diagram of the relative positions of the deformed workpiece and the initially undeformed material is shown in Fig. 1. Axisymmetric conditions are assumed. At time t = 0, the sheet plate lies in the y = 0 plane and is tangential to the punch. The material point with coordinate ((, 0) initially at the middle surface is deformed to (~ + u, w) at time t, where u and w are the displacement in the radial and axial directions respectively. The principal membrane stretch ratios 21 and 22 at the middle surface in the radial and circumferential directions are respectively
71
(2)
In the above, E, Et, v and R are Youngs modulus, the tangent modulus, Poisson's ratio and an anisotropic parameter, respectively. The radial and circumferential components of the Kirchhoff stress are represented by z 1 and z2 ' ze is the effective stress. If Poisson's ratio v is set
T.-C Hsu, C-H. Chu / Journal of Materials Processing Technology 54 (1995) 70-75
72
equal to R/(I + R), Eqs (5)--(7) can be combined as [ ~~21 ] : where
A E~'-] [_T21 '
(9)
[ 1 [E + ~l---\El
A=
Eq. (16), the virtual-work principle in the membrane case can be expressed as ho ~ ('c16e, + "c26e2)dAo - 2ho f ,)Ao dAo
l'~a I~ z
fE
l'~abq
/re
g-1
( E _ l'~b2 | ' 1 + C~\Et }z2
(1o) a
R =
~'1
--
- - ' 1 S 2 ,
I+R R
b=~7 2 - --z,, I+R
(11) (12)
= ~ (716a +Lr~)OAo. dA o
fiffi ] d
+ ZE(~---+--~u)2~Ao
(18)
After using the shape function to discretize the velocity field ti and ~, the general matrix form can be summarized as K tang 0 = i#,
(19)
where
and 1
0
for plastic loading, otherwise.
o/2
Ktang = (13)
The uniaxial stress-strain relationships are assumed to satisfy the following relationship: r =
it¢ffi~+ K~¢6~¢ dAo + ho __.fAo~ ~z'[(l + u¢)2 + w~]
+ r2E2~2)
Ee for z < ry, Ke n for r > Zy
(14)
where K is the strength coefficient, n is the strain-hardening coefficient and ry is the yield stress. The tangent modulus Et is then the slope of the curve dr Et = d-~"
(15)
2.3. Principal of virtual work Since the principal Kirchhoff stress in the radial and circumferential directions are denoted by zl and z2, the virtual-work principle can be written as
f th°/2 (z16~1 + z26~2)&IdAo = I (fxhfi +f26~)dAo, d d - ho/2 (16) where ho is the initial sheet thickness, dAo denotes the undeformed differential area, f1 andfz are the force components per unit undeformed sheet area in the horizontal and vertical direction and 6 denotes variation. The differential area dAo is given by dAo = 2n~ d~.
(17)
I~
NT[HTDH + Zl K(1) + z2K(2)]NdrldAo
J J ho/z
(20)
and P is the node force rate. The nodal velocity vector at ith node is arranged as ~/memb.... = (/,~(i), ff(1))T
(21)
and the line form shape function is used. The exact form of matrix K tangt and ~" can be found in [10].
2.5. Computational algorithm The method implemented in the present work for satisfying the contact and frictional constraint follows an earlier scheme by Kobayashi [2]. Both the stick and slip conditions are considered. The friction stress is governed by the Coulomb friction law. The blank-holding force and the hydroforming pressure in the drawing and hydroforming case are introduced by using the traditional finite-element approach [11] and the resultants occuring at each node are then transformed into the vertical and horizontal directions. The finite-element equations are solved by the Newton-Raphson scheme. Initial boundary conditions and node velocities are assumed based on the previous solution and corrected by the equilibrium check. After the converged solution is obtained, the corresponding boundary condition and contact constraint are used for comparison with the initial assumption. If any of these assumptions are violated, the calculation is repeated until all of the requirements are satisfied. 3. Applications
2.4. Stiffness equations By using the strain-displacement equations (1)(4), constitutive equations (5)(15) and the linearized form of
As a verification of the present formulation and computer program, it has been applied to several cases which were investigated by other authors, and also compared
73
T.-C Hsu, C.-H. Chu /Journal of Materials Processing Technology 54 (1995) 70-75
Table 1 Process
Punch radius (mm)
Stretching Hydroforming Deep drawing (by D.M. Woo)
50.8 25.0 25.4
Die profile radius (mm) 6.35 8.00 12.7
Die throat radius (mm)
Materialthickness (mm)
110 52.4
0.8 0.8
56.39
0.889
Table 2 Process
Material
Stretching, hydroforming BADDQ Deep drawing (by D.M. Woo) Softcopper
E (GPa)
R
Tr (MPa)
K (MPa)
n
250.256
2.2
88.31
548.96
0.23
with the present experimental data. The dimensions of the sheet and tool for these examples are shown in Table 1. The mechanical properties of the sheet, including anisotropic parameters, were measured before the deformation and are list in Table 2. The strain distributions were measured along the two principal directions using the grid method and the experimental data used to compare with the computer simulation results. The examples presented below demonstrate the efficiency of the p r o g r a m in solving some practical sheet metal forming problems. 20 equally-spaced elements are used for all of these simulation cases. Axisymmetric punch stretching experiments were conducted using a hydraulic press. The punch increment was 0.01 m m and lubricated condition was modeled by taking the friction coefficient as a constant value of 0.13. Comparisons of the results of simulation and experimental data are shown in Figs. 2(a) and (b) for radial and circumferential strain, respectively, the solid line being the simulation results and the points the experimental data. Both the radial and circumferential strain distributions show good agreement with the experimental measurements. The experimental data of W o o 1-12] are used here to evaluate the reliability of the present computational scheme for the deep-drawing case. The blank-holding force is 5 kN. The friction coefficient was suggested by W o o to be # = 0.04. The experimental parameters are listed in Tables 1 and 2. Figs 3(a) and (b) show the numerical and experimental results for thickness strain distributions at two values of drawing ratio, the latter being defined as the ratio between the current rim radius and the original rim radius. The results for punch load versus punch travel are shown in Fig. 4. It is seen that both the strain distribution and the punch load obtainedwith the present model agree well with the experimental results.
tr = 82 + 423e°'s°4 (MPa)
0.4
0.3
i
0.2
0.1 Nwnerkal S'imnh~nu ~qx~Dm
•
i - b - | - | - j
0.0
0
10
20
3o
4o
50
60
UndefenaedCcolflln~
03t 0~2"
0• / •_ .
"i o.x.
___~e •
•
@
Punch Travel-- 26.3mm -'~ NumericalSim.l~rlnn •
.
0.0 0
10
Expe~ent Data
.
. 20
. 30
. 40
50
60
Undeformed Coozdin~
Fig. 2. Comparison between numerical and experimental results (stretching): (a) radial strain; (b) thickness strain.
74
T.-C. Hsu, C-H. Chu / Journal of Materials Processing Technology 54 (1995) 70-75
0.1
0.10 Blank Holding Forceffi3.04Ton Hydraulic Pn~ssureffil0MPa Numerical Simulation • ~ n t Data
_m=~-
0.0 0.05
iHmii•
~-0.1
~
I ---I---
~-0.2
Sukhomllnov, Engelsberg and
Davydov [13] [ 12]
expmnent
-0.3
)
10
-
|
-
|
-
i
/ /,,~
-0.05 '
s
20 30 40 Current Coordinate
0.00"1
0.00
•
50 60 (ram)
"
-0.10
•
!
!
I
|
!
10
20
30
40
50
60
Undefonned ~ a t e
-0.1 0.2 Blank Holding Force=3.01ton HydrauSe Prcssure~29.5MPa Numerical Simulation
~-0.3 -0,4
.... •
-0.5 0
. 10
i 0.1
Sukholinov, Engelshetg and Davydov {13} expe~ment [121
.
. . 20 30 CurrentCoordinate
40
50 (ram)
Fig. 3. Effect of drawing ratio on thickness strain: (a) drawing ratio 0.97; (b) drawing ratio 0.83.
40'
presentmethod L,G.Sukhomlinov,V.K.Engelsberg andV.NDavydov[13]
0.0
4).1
i
0
10
|
|
|
!
20 30 40 50 Undefomed Coordinate
60
Fig. 5. Comparison between numerical simulation and experimental data (hydroforming): (a) punch travel 25.9 mm; (b) punch travel 47.1 mm. F(KN) 20'
10"
0
-
0
-
|
~
|
10
20 U(mrn)
30
40
Fig. 4. Hemispherical punch deep drawing: punch versus punch travel•
The hydroforming experiment was conducted using a Amino 130 tonne forming machine. The same B A D D Q steel as for the stretching case was used, the dimensions of the set-up being listed in Table 1. The lubricated condition was modeled by taking p = 0.13. In this process, the punch deforms the blank to its final shape by moving against a controlled pressurized fluid which is different from the drawing process. The pressure path was taken as constant through the process. The thickness strain
T.-C Hsu, C.-H. Chu / Journal of Materials Processing Technology 54 (1995) 70-75
processes such as stretching, drawing and hydroforming, which will be useful for determining the drawing ratio and formability of the sheet.
GI
0~"
!
Acknowledgements The authors wish to thank the National Science Council of Taiwan for the financial support of this research by grant NSC82-0401-E-155-057. The experimental facility provided by China Steel company and Metal Industries Development Centre are also greatly appreciated.
-0.1"
-0.2"
fl/t
,
I 413"
75
tly0mZorming
DrawingRafio~0.83
,,-~wj
References
1.$ Hydraulic Ptessure = ~
MPa
i m t IStep ltan t: ComputationalTTime t, ( ~ n w l l a t i ~ a |
-0.41 0
. 10
.
. 20
. 30
40
I 50
Ctm~nt Cotmfinate Fig. 6. Comparison between deep drawing and hydroforming (Drawing ratio = 0.83)
distributions at two different punch travel length were shown in Figs. 5(a) and (b). The numerical simulation results agree well with the experimental data. The advantage of using the hydroforming process to enforce the formability of the sheet can be seen in Fig. 6 for which the experimental parameters of Woo were used. The thickness strain is reduced by the imposing of hydraulic pressure and the strain distribution is more uniform in the hydroforming process than it is in the conventional drawing case.
4. Conclusions An axisymmetric elastic-plastic finite-element membrane model for the analysis of sheet metal forming processes has been developed. The non-linear problem has taken into account the changes of geometry, material properties and contact conditions during time step. The results presented in this paper show that the computational method can be applied successfully to various
[1] D.M. Woo, The stretching forming test, The Enoineer, 200 (1965) 876-880. [2i S. Kobayashi, S.I. Oh and T. Altan, Metal Forming and the Finite Element Method, Oxford University Press, London, (1989). [3] N.M. Wang and B. Budiansky, Analysis of sheet metal stamping by a finite element method, ASME J. Appl. Mechs., 100 (1978) 73-82. [4] N.M. Wang, Large plastic deformation of a circular sheet caused by a punch stretching, ASME J. Appl. Mechs., 37 (1970) 431-439. [5] N.M. Wang and M.L. Wenner, Effects of strain-hardening 2036T4 representation in sheet metal forming calculation of aluminium, in: J.R. Newby and B.A. Niemerer, (Eds.,) Formability of Metallic Materials-2OOOA.D., ASTM STP 753, ASTM, 1982, pp. 84-104. [6] N.M. Wang and M.L. Wenner, Elastic-Viscoplastic analysis of simple stretch forming problem, Proc. General Motors Research Laboratories Symposium, Plenum Press, New York, 1981, pp. 367-400. [7] S.C. Tang, Large elasto-plastic strain analysis of Flange hole forming, Comput. Struct., 13 (1981) 363-370. [8] A.S. Wifi, An incremental complete solution of the stretch-forming and deep-drawing of a circular blank using a hemispherical punch, Int. J. Mech. Sci., 18 (1976) 23-31. [9] N. Triantafyllidis and S.K. Samanta, Bending effects on flow localization in metallic sheets, Proc. R. Soc. Lond, A 406 (1986) 205-226. [10] N.M. Wang and S.C. Tang, Analysis of bending effects in sheet forming operations, Int. J. Num. Methods En9., 25 (1988) 253-267. [11] O.C. Zienkiewicz, The Finite Element Methods, McGraw-Hill, UK, Ltd 1977. [12] D.M. Woo, On the complete solution of the deep-drawing problem, Int. J. Mech. Sci., 10 (1968) 83. [13] L.G. Sukhomlinov, V.K. Engelsberg and V.N. Davydov, A finite element membrane model for the analysis of axisymmetric sheet metal forming processes, Int. J. Mech. Sci., 34(3) (1992) 179-193.
Materials Processing Technology Journal of Materials Processing Technology 54 (1995) 70-75
ELSEVIER
A finite-element analysis of sheet metal forming processes Tze-Chi Hsu*, Chan-Hung Chu Department of Mechanical Engineering, Yuan-He Institute of Technology, Far East Rd., Tao Yuan, Taiwan 32026, ROC
Received 1 April
Industrial summary
A computational method based on the membrane theory for the analysis of axisymmetricsheet metal forming processes such as punch stretching, deep drawing and hydroforming is presented. The elastic-plastic finite element approach is based on the flow rule associated with Hill's quadratic yield criterion for anisotropic material. The total Lagrangian formulation and virtual work theory are used to derive the stiffnessequations and the relationship between displacementand strain. Some examples on the stretching,drawing and hydroforming of metal sheets are considered,and the computed results are compared with experimental data and with the results from existing numerical solutions.
1. Introduction
One of the first numerical analyses of the sheet metal forming process was done by Woo [1], who considered the material as isotropic and used a finite-difference method to solve the case of axisymmetric punch stretching and drawing. Although his results show good agreement with experimental data in the flange section, a significant discrepancy was observed in the thickness-strain distribution in the tooling/workpiece contact region, this phenomena being attributed to the changing friction conditions. Kobayashi [-2] and his co-workers have worked on several sheet forming problems including drawing, bulging, stretch forming and bending, solving the problems using a finite-element method based on incremental strain theory. In their work, the material was assumed as rigid-plastic (elastic deformation was neglected) and anisotropy and work-hardening characteristics were included. An elastic-plastic F.E.M. formulation of stretch forming for a punch and a die of arbitrary shape was introduced by Wang and Budiansky [3]. The material was governed by the Ramberg-Osgood relationship. Complete boundary conditions covering both the sliding and the sticking condition were provided. Comparisons were made by checking the case of hemispherical punch stretching with rigid-plastic results [-4] where it was noted that the strain distributions were essentially the same until the onset of material unloading in the punch contact region. The explicit form of the finite-element
*Correspondingauthor. 0924-0136/95/$09.50 © 1995Elsevier ScienceS.A. All rights reserved SSDI 0924-0136(95)01922-2
stiffness equations for the hemispherical stretching case was provided by Wang and Wenner [5]. An elastic-viscoplastic formulation which included strain-rate effects was given also by Wang and Wenner [6] and showed that rate sensitivity yields some benefits in punch stretching. Another numerical simulation based on membrane theory was derived by Tang [-7], who used a tensor formulation and applied finite inelastic strain theory to solve a flanged-hole forming problem with four different punch shapes. The results revealed that the strain path during the forming process is not affected by the punch shape but that the maximum punch load depends on the punch shape. All the numerical analysis mentioned above were obtained using simple membrane theory which does not include the variation of stresses through the sheet thickness. The assumption is appropriate in many sheet metal forming operations because the sheet thickness is usually small in comparison with the radius of either the punch or the die. However, in order to simulate bending phenomena such as the early deformation in stretch forming, spring-back etc., inclusion of the bending stress is necessary. Wifi [8] employed an incremental variation method to analyze axisymmetric elastic plastic solids at large strains. The method was applied to the problems of stretch forming and complete deep-drawing of a circular blank using a hemispherical punch. The contact problem in different zones was formulated and tackled by the finite-element method. Although the contact problem at the blank hoder, die, die profile and punch head were included, only a frictionless condition at the tooling/workpiece interface was assumed.
T.-C. Hsu, C.-H. Chu / Journal of Materials Processing Technology 54 (1995) 70-75
Triantafyllidis and Samanta [9] derived a total Lagrangian F.E.M. formulation for large displacements and rotations and large strain deformations of the shell of revolution under axisymmetric loading. The equilibrium equations were derived from the principal of virtual work and a frictionless condition was assumed. Another bending F.E.M. based on the total Lagrangian formulation was derived by Wang and Tang [10], a conventional displacement method being used to construct the principal of virtual work. Finite-element algorithms for axisymmetric and plane-strain cases were presented. The difference between the strain distributions for the membrane and bending results is quite small for axisymmetric punch stretching and cup drawing with typical tool radii. In the present work an analysis of a sheet metal operation involving punch stretching, drawing and hydroforming is developed using the finite-element method. To analyze these processes mathematically, the non-linearities caused by the elastic-plastic material property, and the large strains and displacements, must be considered. The Lagrangian description of motion is applied to the finite-deformation analysis of the thin sheet structure. Only membrane strains are considered. The plane-stress assumption (the stress component normal to the sheet is negligible) is adopted. Since the coordinates used here are the principal directions, only two stress- and strain-components are used in the basic equations. The sheet material is assumed to satisfy a rate insensitive, Mises-type flow rule which takes into account work hardening and anisotropy. At the tool-sheet interface, the friction stress is calculated from the coulomb friction law whenever relative sliding occurs.
o/
Ini
e,< = ln(2,,),
(3)
where ~ = 1, 2. The strain rates are defined as
,~==
-/~
(4)
where • denots the differential with respect to a time-like variable such as punch travel.
2.2. Constitutive equations The model treats plastic flow by a modified von Mises flow rule which satisfy Hill's rate-insensitive normal anisotropic relationships [7]. The strain rates consist of an elastic part and a plastic part, given by
where
~+u
- -
{/~(P'}
~,d° ~
1[1
.~-'/~(lp)~
,.~ee,
(5)
--v][lz~ ' Et
1
(6)
1 -I-- R
"el
1VR and
2R
~1/2 •
(8)
(1)
and
22 =
X
where subscript ¢ represents partial differential with respect to the initial radial position, ~. The principal logarithmic strains el and e2, in the radial and circumferential directions are respectively,
2.1. Displacement and strain equations
~lZ
u
Fig. 1. Initial and current positions of material in a shell.
{e} = {e~'~} + {e'~},
& = [(1 + u¢) 2 + w~]
Cz
%
2. Finite element formulation
A schematic diagram of the relative positions of the deformed workpiece and the initially undeformed material is shown in Fig. 1. Axisymmetric conditions are assumed. At time t = 0, the sheet plate lies in the y = 0 plane and is tangential to the punch. The material point with coordinate ((, 0) initially at the middle surface is deformed to (~ + u, w) at time t, where u and w are the displacement in the radial and axial directions respectively. The principal membrane stretch ratios 21 and 22 at the middle surface in the radial and circumferential directions are respectively
71
(2)
In the above, E, Et, v and R are Youngs modulus, the tangent modulus, Poisson's ratio and an anisotropic parameter, respectively. The radial and circumferential components of the Kirchhoff stress are represented by z 1 and z2 ' ze is the effective stress. If Poisson's ratio v is set
T.-C Hsu, C-H. Chu / Journal of Materials Processing Technology 54 (1995) 70-75
72
equal to R/(I + R), Eqs (5)--(7) can be combined as [ ~~21 ] : where
A E~'-] [_T21 '
(9)
[ 1 [E + ~l---\El
A=
Eq. (16), the virtual-work principle in the membrane case can be expressed as ho ~ ('c16e, + "c26e2)dAo - 2ho f ,)Ao dAo
l'~a I~ z
fE
l'~abq
/re
g-1
( E _ l'~b2 | ' 1 + C~\Et }z2
(1o) a
R =
~'1
--
- - ' 1 S 2 ,
I+R R
b=~7 2 - --z,, I+R
(11) (12)
= ~ (716a +Lr~)OAo. dA o
fiffi ] d
+ ZE(~---+--~u)2~Ao
(18)
After using the shape function to discretize the velocity field ti and ~, the general matrix form can be summarized as K tang 0 = i#,
(19)
where
and 1
0
for plastic loading, otherwise.
o/2
Ktang = (13)
The uniaxial stress-strain relationships are assumed to satisfy the following relationship: r =
it¢ffi~+ K~¢6~¢ dAo + ho __.fAo~ ~z'[(l + u¢)2 + w~]
+ r2E2~2)
Ee for z < ry, Ke n for r > Zy
(14)
where K is the strength coefficient, n is the strain-hardening coefficient and ry is the yield stress. The tangent modulus Et is then the slope of the curve dr Et = d-~"
(15)
2.3. Principal of virtual work Since the principal Kirchhoff stress in the radial and circumferential directions are denoted by zl and z2, the virtual-work principle can be written as
f th°/2 (z16~1 + z26~2)&IdAo = I (fxhfi +f26~)dAo, d d - ho/2 (16) where ho is the initial sheet thickness, dAo denotes the undeformed differential area, f1 andfz are the force components per unit undeformed sheet area in the horizontal and vertical direction and 6 denotes variation. The differential area dAo is given by dAo = 2n~ d~.
(17)
I~
NT[HTDH + Zl K(1) + z2K(2)]NdrldAo
J J ho/z
(20)
and P is the node force rate. The nodal velocity vector at ith node is arranged as ~/memb.... = (/,~(i), ff(1))T
(21)
and the line form shape function is used. The exact form of matrix K tangt and ~" can be found in [10].
2.5. Computational algorithm The method implemented in the present work for satisfying the contact and frictional constraint follows an earlier scheme by Kobayashi [2]. Both the stick and slip conditions are considered. The friction stress is governed by the Coulomb friction law. The blank-holding force and the hydroforming pressure in the drawing and hydroforming case are introduced by using the traditional finite-element approach [11] and the resultants occuring at each node are then transformed into the vertical and horizontal directions. The finite-element equations are solved by the Newton-Raphson scheme. Initial boundary conditions and node velocities are assumed based on the previous solution and corrected by the equilibrium check. After the converged solution is obtained, the corresponding boundary condition and contact constraint are used for comparison with the initial assumption. If any of these assumptions are violated, the calculation is repeated until all of the requirements are satisfied. 3. Applications
2.4. Stiffness equations By using the strain-displacement equations (1)(4), constitutive equations (5)(15) and the linearized form of
As a verification of the present formulation and computer program, it has been applied to several cases which were investigated by other authors, and also compared
73
T.-C Hsu, C.-H. Chu /Journal of Materials Processing Technology 54 (1995) 70-75
Table 1 Process
Punch radius (mm)
Stretching Hydroforming Deep drawing (by D.M. Woo)
50.8 25.0 25.4
Die profile radius (mm) 6.35 8.00 12.7
Die throat radius (mm)
Materialthickness (mm)
110 52.4
0.8 0.8
56.39
0.889
Table 2 Process
Material
Stretching, hydroforming BADDQ Deep drawing (by D.M. Woo) Softcopper
E (GPa)
R
Tr (MPa)
K (MPa)
n
250.256
2.2
88.31
548.96
0.23
with the present experimental data. The dimensions of the sheet and tool for these examples are shown in Table 1. The mechanical properties of the sheet, including anisotropic parameters, were measured before the deformation and are list in Table 2. The strain distributions were measured along the two principal directions using the grid method and the experimental data used to compare with the computer simulation results. The examples presented below demonstrate the efficiency of the p r o g r a m in solving some practical sheet metal forming problems. 20 equally-spaced elements are used for all of these simulation cases. Axisymmetric punch stretching experiments were conducted using a hydraulic press. The punch increment was 0.01 m m and lubricated condition was modeled by taking the friction coefficient as a constant value of 0.13. Comparisons of the results of simulation and experimental data are shown in Figs. 2(a) and (b) for radial and circumferential strain, respectively, the solid line being the simulation results and the points the experimental data. Both the radial and circumferential strain distributions show good agreement with the experimental measurements. The experimental data of W o o 1-12] are used here to evaluate the reliability of the present computational scheme for the deep-drawing case. The blank-holding force is 5 kN. The friction coefficient was suggested by W o o to be # = 0.04. The experimental parameters are listed in Tables 1 and 2. Figs 3(a) and (b) show the numerical and experimental results for thickness strain distributions at two values of drawing ratio, the latter being defined as the ratio between the current rim radius and the original rim radius. The results for punch load versus punch travel are shown in Fig. 4. It is seen that both the strain distribution and the punch load obtainedwith the present model agree well with the experimental results.
tr = 82 + 423e°'s°4 (MPa)
0.4
0.3
i
0.2
0.1 Nwnerkal S'imnh~nu ~qx~Dm
•
i - b - | - | - j
0.0
0
10
20
3o
4o
50
60
UndefenaedCcolflln~
03t 0~2"
0• / •_ .
"i o.x.
___~e •
•
@
Punch Travel-- 26.3mm -'~ NumericalSim.l~rlnn •
.
0.0 0
10
Expe~ent Data
.
. 20
. 30
. 40
50
60
Undeformed Coozdin~
Fig. 2. Comparison between numerical and experimental results (stretching): (a) radial strain; (b) thickness strain.
74
T.-C. Hsu, C-H. Chu / Journal of Materials Processing Technology 54 (1995) 70-75
0.1
0.10 Blank Holding Forceffi3.04Ton Hydraulic Pn~ssureffil0MPa Numerical Simulation • ~ n t Data
_m=~-
0.0 0.05
iHmii•
~-0.1
~
I ---I---
~-0.2
Sukhomllnov, Engelsberg and
Davydov [13] [ 12]
expmnent
-0.3
)
10
-
|
-
|
-
i
/ /,,~
-0.05 '
s
20 30 40 Current Coordinate
0.00"1
0.00
•
50 60 (ram)
"
-0.10
•
!
!
I
|
!
10
20
30
40
50
60
Undefonned ~ a t e
-0.1 0.2 Blank Holding Force=3.01ton HydrauSe Prcssure~29.5MPa Numerical Simulation
~-0.3 -0,4
.... •
-0.5 0
. 10
i 0.1
Sukholinov, Engelshetg and Davydov {13} expe~ment [121
.
. . 20 30 CurrentCoordinate
40
50 (ram)
Fig. 3. Effect of drawing ratio on thickness strain: (a) drawing ratio 0.97; (b) drawing ratio 0.83.
40'
presentmethod L,G.Sukhomlinov,V.K.Engelsberg andV.NDavydov[13]
0.0
4).1
i
0
10
|
|
|
!
20 30 40 50 Undefomed Coordinate
60
Fig. 5. Comparison between numerical simulation and experimental data (hydroforming): (a) punch travel 25.9 mm; (b) punch travel 47.1 mm. F(KN) 20'
10"
0
-
0
-
|
~
|
10
20 U(mrn)
30
40
Fig. 4. Hemispherical punch deep drawing: punch versus punch travel•
The hydroforming experiment was conducted using a Amino 130 tonne forming machine. The same B A D D Q steel as for the stretching case was used, the dimensions of the set-up being listed in Table 1. The lubricated condition was modeled by taking p = 0.13. In this process, the punch deforms the blank to its final shape by moving against a controlled pressurized fluid which is different from the drawing process. The pressure path was taken as constant through the process. The thickness strain
T.-C Hsu, C.-H. Chu / Journal of Materials Processing Technology 54 (1995) 70-75
processes such as stretching, drawing and hydroforming, which will be useful for determining the drawing ratio and formability of the sheet.
GI
0~"
!
Acknowledgements The authors wish to thank the National Science Council of Taiwan for the financial support of this research by grant NSC82-0401-E-155-057. The experimental facility provided by China Steel company and Metal Industries Development Centre are also greatly appreciated.
-0.1"
-0.2"
fl/t
,
I 413"
75
tly0mZorming
DrawingRafio~0.83
,,-~wj
References
1.$ Hydraulic Ptessure = ~
MPa
i m t IStep ltan t: ComputationalTTime t, ( ~ n w l l a t i ~ a |
-0.41 0
. 10
.
. 20
. 30
40
I 50
Ctm~nt Cotmfinate Fig. 6. Comparison between deep drawing and hydroforming (Drawing ratio = 0.83)
distributions at two different punch travel length were shown in Figs. 5(a) and (b). The numerical simulation results agree well with the experimental data. The advantage of using the hydroforming process to enforce the formability of the sheet can be seen in Fig. 6 for which the experimental parameters of Woo were used. The thickness strain is reduced by the imposing of hydraulic pressure and the strain distribution is more uniform in the hydroforming process than it is in the conventional drawing case.
4. Conclusions An axisymmetric elastic-plastic finite-element membrane model for the analysis of sheet metal forming processes has been developed. The non-linear problem has taken into account the changes of geometry, material properties and contact conditions during time step. The results presented in this paper show that the computational method can be applied successfully to various
[1] D.M. Woo, The stretching forming test, The Enoineer, 200 (1965) 876-880. [2i S. Kobayashi, S.I. Oh and T. Altan, Metal Forming and the Finite Element Method, Oxford University Press, London, (1989). [3] N.M. Wang and B. Budiansky, Analysis of sheet metal stamping by a finite element method, ASME J. Appl. Mechs., 100 (1978) 73-82. [4] N.M. Wang, Large plastic deformation of a circular sheet caused by a punch stretching, ASME J. Appl. Mechs., 37 (1970) 431-439. [5] N.M. Wang and M.L. Wenner, Effects of strain-hardening 2036T4 representation in sheet metal forming calculation of aluminium, in: J.R. Newby and B.A. Niemerer, (Eds.,) Formability of Metallic Materials-2OOOA.D., ASTM STP 753, ASTM, 1982, pp. 84-104. [6] N.M. Wang and M.L. Wenner, Elastic-Viscoplastic analysis of simple stretch forming problem, Proc. General Motors Research Laboratories Symposium, Plenum Press, New York, 1981, pp. 367-400. [7] S.C. Tang, Large elasto-plastic strain analysis of Flange hole forming, Comput. Struct., 13 (1981) 363-370. [8] A.S. Wifi, An incremental complete solution of the stretch-forming and deep-drawing of a circular blank using a hemispherical punch, Int. J. Mech. Sci., 18 (1976) 23-31. [9] N. Triantafyllidis and S.K. Samanta, Bending effects on flow localization in metallic sheets, Proc. R. Soc. Lond, A 406 (1986) 205-226. [10] N.M. Wang and S.C. Tang, Analysis of bending effects in sheet forming operations, Int. J. Num. Methods En9., 25 (1988) 253-267. [11] O.C. Zienkiewicz, The Finite Element Methods, McGraw-Hill, UK, Ltd 1977. [12] D.M. Woo, On the complete solution of the deep-drawing problem, Int. J. Mech. Sci., 10 (1968) 83. [13] L.G. Sukhomlinov, V.K. Engelsberg and V.N. Davydov, A finite element membrane model for the analysis of axisymmetric sheet metal forming processes, Int. J. Mech. Sci., 34(3) (1992) 179-193.