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Blank design and strain estimates for sheet metal forming processes by a finite element inverse approach with initial guess of linear deformation
Journal of Materials Processing Technology 82 (1998) 145 – 155
Blank design and strain estimates for sheet metal forming processes by a finite element inverse approach with initial guess of linear deformation C.H. Lee, H. Huh * Department of Mechanical Engineering, Korea Ad6anced Institute of Science and Technology, 373 -1 Kusongdong, Yusonggu, Taejon 305 -701, South Korea Received 20 March 1997
Abstract A new finite element approach is introduced for direct prediction of blank shapes and strain distributions from desired final shapes in sheet metal forming. The approach deals with the geometric compatibility of finite elements, plastic deformation theory, minimization of plastic work with constraints and a proper initial guess. The algorithm developed is applied to cylindrical cup drawing, square cup drawing and oil pan drawing in order to confirm its validity by demonstrating reasonably accurate numerical results for each problem. Rapid calculation with this algorithm enables easy determination of various process variables for design of a sheet metal forming process. © 1998 Elsevier Science S.A. All rights reserved. Keywords: Blank design; Finite element inverse approach; Linear deformation
1. Introduction Sheet metal forming processes experiences very complicated deformation effected by process parameters such as the die geometry, the blank shape, the sheet thickness, the blank holding force, friction, lubrication and so on. Although these process parameters have an influence on the deformation mechanism and the quality of deformed parts, the optimum condition for process parameters is determined by intuition and experience, with trial-and-error. More systematic methods for the determination of the optimum conditions have been developed with the aid of computers and numerical analysis. Numerical analysis including finiteelement methods, can simulate complicated sheet metal parts and affords useful information on forming processes, thus reducing trial-and-error. However, numerical simulation is generally carried out with given process parameters and so requires numerical trial-anderror with enormous time and cost, to determine the optimum process parameters. For this reason, there * Corresponding author. Fax: +82 42 8693210/8695210. 0924-0136/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S0924-0136(98)00034-X
arises the necessity for some approaches to find one, or some, of the optimum process parameters directly. One of the important process parameters is the blank shape, which has a direct relationship with the quality of the deformed parts. Design methods for the blank shape have been widely studied by many researchers. Jimma [1], Hazek and Lange [2] and Karima [3] made use of the slip-line method to design the initial blank shape. Vogel and Lee [4] and Chen and Sowerby [5] used the characteristic of plane stress, while Duncan et al. [6] and Blount and Stevens [7] used geometric mapping to design the initial blank shape. These methods provide good guidance to design the initial blank shape, even though they neglect the height of the deformed parts, have geometric restrictions, or do not consider the deformation behavior of the materials. On the other hand, there have been several attempts to design the blank shape and estimate the distribution of the strain in a deformed part with deformation theory. Majlessi and Lee [8–10] extended the theory of Levy et al. [11] and applied it to axisymmetric problems and axisymmetric multi-stage problems, obtaining good results. Guo and Batoz [12–15] derived a formulation for field problems as an inverse method to obtain the initial
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tion theory and Hill’s anisotropic yield criterion. The calculation and minimization adopts an initial guess using inverse mapping with linear deformation, which is of great importance for good convergence. Minimization provides the optimum shape of the initial blank and the strain distribution in a deformed part under given friction conditions and blank-holding forces. The preapproximated inverse method enables the determination of process parameters within a small range of error with low computing time, affording the determination of the strain distribution in a deformed part before the process design.
2. Formulation Fig. 1. Kinematics of linear triangular membrane elements between the initial state and the final state.
blank shape and the thickness distribution in a deformed part. Chung and Richmond [16 – 19] suggested ideal forming with optimum deformation to design the initial blank shape and the intermediate deformed shapes. In this paper, the strain tensor at the deformed state is calculated from the initial state as a function of the coordinates by approximating the deformed shape to a system of discretized triangular membrane elements. The plastic work is then calculated element-wise and minimized satisfying the given constraints with the conjugate gradient method and the Newton – Raphson method. The plastic work is assumed to obey Hencky’s deforma-
Predicting an initial blank shape from a final deformed shape in a one-step calculation, the finite element approach using deformation theory has some different features from the conventional finite element method using the incremental theory. The problem reduces to minimizing the plastic potential energy, relating the initial state to the final state. In the problem, the given variables are the geometry of the final state and the thickness of the initial state, while the unknowns are the coordinates of the initial state and the thickness of the final state. These unknowns are determined by minimization of the objective function, which is an approximated plastic potential energy derived from the plastic work and the external work as a function of the unknowns.
Fig. 2. Transformation of elements from the final state to the initial state.
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147
Fig. 3. Deformed shapes of cylindrical cups with finite-element mesh geometry obtained from CAD models: (a) cup height =20 mm; (b) cup height =30 mm; and (c) cup height=40 mm.
2.1. Plastic potential
thoroughly by various numerical examples and corresponding experiments.
A deformable body in equilibrium satisfies the principle of virtual work as the necessary and sufficient condition for the stress field. Hill assumed the existance of a plastic potential that has its minimum value when a plastic body is in equilibrium [20]: min. C =
&
D
s:o; dV −
&
(D
t · 7 dG
(1)
The above equation holds in any bodies in the plastic state with large deformation, regardless of their constitutive relationships. In this formulation, the plastic potential energy in the final state is derived as a function of the initial coordinates, neglecting the intermediate deformation path, which might lead to absurd results with remarkable errors. This kind of analysis therefore, must be modulated carefully and examined
2.2. Kinematics The geometry of the initial and final state is discretized by linear triangular membrane elements. Fig. 1 shows the schematic elements of the initial and final states. By the assumption of proportional loading and the minimum plastic work path [19,26], the strain can be identified as the logarithmic strain which ensures the incompressibility of the material during large deformation. The logarithmic strain components oi (X) are expressed as Eq. (2) from the Green deformation tensor [21]:
Æo xx Ç Æln l 1 cos2 u+ ln l2 sin2 u Ç [oi (X)] = à o yy à = Ãln l 1 sin2 u+ ln l2 cos2 u à Èo xy É È(ln l 1 − ln l2) sin u cos u É (2) 6where l1, l2 are defined by the principal values of the Green deformation tensor and u designates the direction of l1 with respect to the X coordinates.
Fig. 4. Finite-element meshes for an initial guess and the computed blank shape of a cylindrical cup: (a) initially-guessed shape by linear inverse mapping; and (b) computed blank shape (cup height = 30 mm).
Fig. 5. Computed blank shapes for various cup heights in cylindrical cup drawing.
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148
Fig. 6. A photograph of cylindrical cups after deep drawing: (a) cup height =20 mm; (b) cup height=30 mm; and (c) cup height= 40 mm.
' '
n n
Æ (C + C ) (C1 −C2) 2 1 2 + +C 23 Ã 2 2 Æl 1 Ç Ã (C 1 + C2) (C1 −C2) 2 − Ãl 2 Ã = Ã +C 23 2 2 Èu É Ã l 1 −C1 Ã tan − 1 C3 È
Ç Ã 1/2Ã Ã Ã Ã É
1/2
(3) The Green deformation tensor can be constructed by two pairs of vectors in Fig. 1 as follows:
The logarithmic strain components in Eq. (1) are constant in a linear triangular element. Since the coordinates of the nodal points in the final state, x, are known, the strain components become functions of the coordinates of the nodal points at the initial state, X. The coordinates X are updated at every iteration of the minimization process.
2.3. Constituti6e relationship The stress–strain rate relationship, which is derived from the associated flow rule and Hill’s anisotropic
[C11 C22 C12]T [C1 C2 C3]T 12G3y +13G2y − (12 + 13 − 11)G2y G3y Æ Ç 12G3x 2 +13G2x 2 − (12 + 13 − 11)G2x G3x à à 1 = à à (G3yG2x −G3xG2y )2 1 + 13 − 11 à −12G3x G3y −13G2x G2y + 2 (G2x G2y + G2y G3x )à 2 È É 2
2
where 1i =g 2ix +g 2iy +g 2iz.
(4)
yield criterion for plane stress, is integrated from the initial state to the final state. Then, the integrated relation is expressed as Eq. (5) in an inverted form. The concept of Eq. (5) is from the Hencky’s deformation theory, i.e. the strain components in the final state are assumed to be proportional to the stress components in the final state. Although the deformation theory should be restricted to infinitesimal deformation, it is assumed that the deformation follows the proportional loading path, allowing errors from the assumption of proportional loading:
Æs xx Ç ÆAB A 0 ÇÆo xx Ç 2 s¯ à s yy à = à A AC 0 Ãà o yy à Ès xy É 3 o¯ È 0 0 ADÉÈo xy É
(5)
where
Fig. 7. Comparison of the flange contours in the final shape between those of the desired CAD models and the experimental contours.
A=
r(2+ r) , (1+ 2r)
D=
1 r
B=
(1+ r) , r
C=
(1+r) , r (6)
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149
Fig. 8. Deformed shapes of square cups with finite-element mesh geometry obtained from CAD models: (a) cup height = 20 mm; (b) cup height =30 mm; and (c) cup height=40 mm.
The effective strain corresponding to Eq. (5) can be expressed as: o¯ =
'
2 A[Bo 2xx + 2oxxoyy +Co 2yy +2Do 2xy] 3
(7)
E
Wp(X) = % e=1
& & De
ef 0
2 s¯ D o do t dA 3 o¯ ab a b
(9)
Eq. (9) shows the plastic work, Wp(X), is also a function of the nodal coordinates, X, at the initial state.
where r in Eq. (6) is the Lankford value.
2.5. Boundary conditions
2.4. Plastic deformation energy
Sheet metal has a relatively large surface area compared to its volume. While the boundary conditions applied to a sheet surface change continuously during deformation, the deformation theory used in the present analysis retains a rigorous modeling of the boundary conditions. For this reason, a simplified method which considers the friction force, the blank-holding force and the draw-bead force, is proposed as follows.
When the domain of the final state is discretized by finite elements, the sum of the plastic deformation energy for each element can be calculated by use of Eq. (8): E
Wp(X) = % e=1
& & De
ef
s:do dV
(8)
0
where the stress and strain components are defined by Eqs. (2) and (5), respectively and of, is the strain tensor of the final state. Since a linear triangular element is adopted, the thickness, as well as the stress and strain components, are constant within an element. Thus, Eq. (8) is simply rewritten as Eq. (9):
2.5.1. Friction boundary conditions The friction force induced by the normal force is considered for the punch and die interface surface. The normal force Fn at a node can be obtained from the equilibrium of sheet metal:
Fig. 9. Finite-element meshes for an initial guess and the computed blank shape of a square cup: (a) initially-guessed shape by linear inverse mapping; and (b) computed blank shape (cup height = 30 mm).
Fig. 10. Four shapes of finite element meshes for an initial guess: (a) by projection; (b) by linear mapping with a scale factor of 0.75; (c) by linear mapping with a circular contour; and (d) by linear mapping with a scale factor of 1.15 (cup height= 30 mm).
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150
angle at the node for each element and the reaction force at the node for each element; nf is the outward unit normal vector at the final state. The normal force in Eq. (10) is an averaged value at the nodal point and supposed to exist where the sheet metal is in bending. The frictional work is then written as follows:
& & n uf
E
Wf(X) = %
e=1
(De
tf du dG
0
= − % mFn tanh a node
U U U
U U
(12)
where tf, m and U are the frictional stress, the friction coefficient and the tangential displacement; uf is the total displacement from the initial state to the final state and U is obtained from Eq. (13): U(X) =[(x − X) −(ni · (x− X))ni]
Fig. 11. Computed blank shapes from four initial guesses: (a) by projection; (b) by linear mapping with a scale factor of 0.75; (c) by linear mapping with circular contour; and (d) by linear mapping with a scale factor of 1.15 (cup height= 30 mm).
(13)
The hyperbolic tangent function in Eq. (12) stabilizes the numerical oscillation when a node in contact becomes almost stuck [22]. The coefficient a is a control number for good convergence, ranging from 1 to approximately 10. ni is the unit normal vector at the initial state.
m
Fn =
*
% uk F k k=1 m
% uk F k
k=1
where Fk =
&
*
(De
· nf
t dG =
(10)
&
s: De
(o dV (u
(11)
In the above equation, m, uk and Fk are the total number of elements that contain a designated node, the
2.5.2. Blank-holding force and draw-bead force The blank-holding force prevents the sheet metal from wrinkling and controls the drawing of sheet metal from the flange. According to Chung and Swift’s experiment [23], most of the blank-holding force is concentrated at the outer edge of the flange where the sheet metal thickens. Thus, the frictional work by the blankholding force can be evaluated at the nodes in the outer edge of a blank sheet:
& & n uf
Eb
Wb(X) = %
e=1
(De
tb du dG
0
= − % mFb tanh a node
U U U
U U
(14)
where Fb is the blank-holding force converted into the nodal values at the outer edge. The work performed by the draw-bead force can be evaluated with the same concept as for the blank holding force:
& & n uf
Ed
Wd(X) = %
e=1
(De
td du dG
0
= − % mFd tanh a node
Fig. 12. Computed blank shapes for various cup heights in square cup drawing.
U U U
U U
(15)
where the draw-bead force Fd can be obtained from either experiment or numerical simulation. The blankholding force, the draw-bead force and the location of the draw-bead can be regarded as process variables determined at the design stage by the present simulation algorithm.
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151
Fig. 13. A photograph of square cups after deep drawing: (a) cup height =20 mm; (b) cup height= 30 mm; and (c) cup height=40 mm.
2.6. Minimization of the objecti6e function
imated plastic potential energy, as follows:
An objective function derived from the plastic potential energy can be minimized to solve inverse problems in sheet metal forming processes. However, the problems do not provide the prescribed static and kinematic boundary conditions. Without incremental steps which follow deformation paths, the punch force and the intermediate deformed shape cannot be calculated and the plastic potential energy in any stage is almost impossible to obtain. In order to overcome these difficulties, an approximated plastic potential energy is defined by the difference between the plastic deformation energy and the equivalent external work performed. The equivalent external work can be obtained from the force equilibrium equation: Fp + Ff +Fb + Fd = 0
or
Fp = −Ff −Fb −Fd
(16)
where Fp is the punch force, Ff is the friction force due to bending, Fb is the blank-holding force and Fd is the draw-bead force. The concept of force equilibrium and the principle of virtual work enables calculation of the so-called equivalent external work and thus, an approx-
min. C(X) Wp(X) − Wf(X) − Wb(X) − Wd(X)
where, Wp(X) is the plastic deformation energy; Wf(X) is the work done by friction due to bending; Wb(X) is the work done by the blank-holding force; and Wd(X) is the work done by the draw-bead force. The formulation for the minimum of C(X) is derived to construct a matrix equation for the conjugate gradient method or the Newton–Raphson method. The local minimum of C(X) can be satisfied when the first derivative of C(X) has a stationary value. The first derivative is expressed as follows: min. C(X)UR(X)
(C(X) =0 (X
(18)
where
&
R(X) E
= % e=1
− %
s:
&
De
Eb
e=1
Fig. 14. Comparison of the flange contours in the final shape between those of the desired CAD models and the experimental contours.
(17)
E (o dV − % (X e=1
(De
&
tb
&
(De
(u dG − % (X e=1 Ed
tf
(u dG (X
(De
td
(u dG (X
(19)
Fig. 15. Comparison of the thickness – strain distribution between computed results and experimental results for square cups (cup height=30 mm).
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152
Fig. 16. Deformed shape of the oil pan with the finite-element mesh geometry obtained from the CAD model.
for the effective stress – strain relationship s¯ = k(o0 + o¯ )n. The conjugate gradient method requires only the first derivatives in Eq. (19). However, the Newton– Raphson method requires the second derivatives and is expressed as follows:
(R(X) (X
n
{dX} = − {R(X)}n
(20)
(n)
X(n + 1) =X(n) +b dX where E (R(X) = % (X e=1
& & De
E
Eb
− % e=1
&
− %
e=1
(21)
(s (o ( 2o : + s: dV (X (X (X2
(tf (u dG (X (X (De
Ed (tb (u dG − % e=1 (De (X (X
&
(td (u dG (De (X (X
Fig. 18. Thickness – strain distributions of the oil pan across line A1 – A2 and line B1 – B2.
(22)
where b is the deceleration coefficient ranging from 0 to 1. The first and second derivatives of the strain components in Eqs. (19) and (22) are derived as:
3 3 4 (ep (ep (la (Cb (Gg = % % % (Xk a = 1 b = 1 g = 1 (la (Cb (Gg (Xk
(p=1− 3; ( 2op ( = (Xj(Xk (Xj
k= 1−6) (ep lla (Cb (Gg a = 1 b = 1 g = 1 (la (Cb (Gg (Xk 3
3
%
4
% %
n
( 2op (la (Cb (Gg r = 1 s = 1 t = 1 a = 1 b = 1 g = 1 (lr (la (Cb (Gg (Xk 3
3
4
3
3
= % % % % × × ×
4
% %
(23)
3 3 4 3 4 (lr (Cs (Gt (op ( 2la + % % % % % (Cs (Gt (Xj a = 1 r = 1 s = 1 b = 1 g = 1 (la (Cr (Cb
(Cb (Gr (Gr (Xk
3 3 4 4 (Cr (Gs + % % % % (Gs (Xj a=1 b=1 r=1 g=1
(op (la ( 2Cb (Gg (Gr (la (Cb (Gr (Gg (Xk (Xj (oi (la (Cb ( 2Gg a = 1 b = 1 g = 1 (la (Cb (Gg (Xk (Xj 3
+ % Fig. 17. Computed blank shape for the oil pan from the CAD model.
3
4
% %
(p=1 −3;
j,k =1− 6)
(24)
C.H. Lee, H. Huh / Journal of Materials Processing Technology 82 (1998) 145–155
The conjugate gradient method and the Newton– Raphson iteration start with an initial guess of a point set X0 obtained in the next section.
2.7. Initial guess by in6erse mapping with linear deformation
E
% e=1
&
E
[B]T[D][B] dV{U} = % Ve
e=1
&
[B]T[D][o0] dV Ve
to the drawing of an oil pan, which has more complicated geometries. The material properties and the process variables are shown below for simulation and experiment into cylindrical and square cup drawing. Stress–strain curve
The non-linearity that comes from the large deformation and the material properties makes it difficult to solve Eq. (17), which essentially needs an initial guess that determines appropriate points X0 before solving with the conjugate gradient method or the Newton– Raphson method. The idea of the initial guess used in the present analysis is a linear mapping of the deformed surface of the final state to a flat surface of the initial state. The mapping procedure discretizes the surface into triangular finite elements and assumes that the material in each element deforms linearly. In order to obtain approximate mapping from the deformed surface to an initial flat surface, two types of transformation between the final and initial state are introduced (Fig. 2). The scheme for an initial guess is as follows. Step 1. Carry out two types of transformation for each element from the final state to the initial state, as shown in Fig. 1. Calculate initial strain o0 between the two transformed elements and assume that an initial strain yields the internal stress. Step 2. Solve the elastic recovery using an elastic finite-element method expressed in Eq. (25) for the elements of‘ the second transformation:
(25)
Step 3. If the norm of the right-hand side of Eq. (25) is less than unity, go to Step 4. Otherwise, update the coordinates and go back to Step 2. Step 4. Modulate the area of the calculated shape as the same area of the final state. This modulation is performed because the initial guess scheme is a reverse process to the real deformation process. The modulated shape is used as an initial guess of a point set X0.
3. Numerical results The algorithm described above has been implemented in a finite-element code and applied to several sheet metal forming examples for validation. The blank shapes and thickness distributions of a cylindrical and a square cup have been obtained as a bench-mark test. In order to check the validity of the present algorithm, experiments have been carried out for cylindrical and square cup drawing with blank specimens prepared for the calculated blank shape. As a demonstration of its versatility, the present algorithm has also been applied
153
s¯ = 54.5 (0.00436+o¯ )0.263 (kgf mm−2) r= 1.87 t= 0.8 mm
Lankford value Initial sheet thickness Friction coefficient m= 0.15 Blank-holding force Fb = 4000 kgf
3.1. Deep drawing of a cylindrical cup The present algorithm is first applied to the deep drawing of a cylindrical cup for demonstration of its validity. The geometries of cylindrical cups discretized with finite elements are shown in Fig. 3. The height of the cylindrical cups are 20, 30 and 40 mm, respectively. The diameter of the cups is 60 mm and the width of the flange is 5 mm. For the sake of symmetry, only one quarter of the cup is considered in the calculation. Fig. 4(a) is an initial guess obtained from inverse mapping with linear deformation and Fig. 4(b) is a blank shape calculated in the analysis from Fig. 4(a) as an initial guess. Both figures show roundness on the outside rim, which demonstrates the stable-mesh independency of the present algorithm. The initial blank shapes for the three cup heights are depicted in Fig. 5, showing that the size of the initial blank increases with the cup height. Blank shapes for normal anisotropic sheet metal are calculated as near-circles. Deep-drawing experiments were performed with the specimens of the calculated blank shapes in Fig. 5, with Fig. 6 presenting a photograph of the cylindrical cups formed in the experiments. The flanges of the deepdrawn cups in the experiments are compared with the final shapes, in the analysis shown in Fig. 7. The difference in Fig. 7 can be explained both by simulation error and experimental error. The simulation error is mainly due to the use of deformation theory, neglecting the anisotropy of the sheet metal and the boundary conditions with simplified equations, while the experimental error is mainly due to the eccentricity of the specimen placement and discrepancy in the distribution of the blank-holding forces. Although the present method calculates as a one-step procedure and has many problems, the error is found to be relatively small, demonstrating that the present method can predict a blank shape within a small range of error.
3.2. Deep drawing of a square cup Square cups have been considered in order to obtain their blank shapes and thickness strain distributions.
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The geometries of the square cups are discretized with finite elements, as shown in Fig. 8. Fig. 9(a) is an initial guess obtained from inverse mapping with linear deformation and Fig. 9(b) is a blank shape calculated in the analysis from Fig. 9(a) as an initial guess. In order to demonstrate the independency of an initial guess, four arbitrary initial guesses, as shown in Fig. 10, are used to calculate the initial blank shape. Fig. 11 fully demonstrates the independency of an initial guess with the present algorithm, since the initial blank shapes obtained are almost the same, although the initial guesses are very different from each other. The initial blank shapes for three cup heights are depicted in Fig. 12. Experiments have been carried out with the calculated blank shapes in Fig. 12, the photographs of the square cups formed in the experiment being shown in Fig. 13. The flange shapes obtained from deep drawing of the initial blanks in experiments are compared with those of CAD models in Fig. 8. The comparison presented in Fig. 14 shows that there are differences in the flange shape between theat desired and that obtained by experiment using calculated blanks, there being an excess in the 45° direction and deficiency in the rolling direction. This is due to the fact that the present algorithm does not take account of the concentration of the blank-holding force in the region where sheet metal becomes thicker during deformation. The thickness-strain distributions along the transverse and diagonal direction of a square cup are shown in Fig. 15. The thickness-strain distributions predicted by the present method are in moderate agreement with the experimental results in the upper region and the flange region of a cup, except for the region near to the wall. Since the incremental finite-element method also calculates the different thickness-strain distribution from experimental results [22], this result obtained with one-step calculation could be acceptable.
3.3. Deep drawing of an oil pan The present algorithm was applied to a more complicated problem of oil-pan drawing for demonstration of its capability and versatility. An oil pan for under an automobile engine is shown in Fig. 16, with finite-element descritization. The geometry of the oil pan is discretized by 1460 elements and 766 nodes. The oil pan is usually manufactured by two steps of drawing in a press shop and needs careful process design. This process has been simulated with incremental finite-element analysis by several researchers [24]. Liu and Karima [25] calculated the blank shape for an oil pan by applying infinitesimal strain equation to a large deformation problem and solving the problem in one step. Regardless of how complicated the process is, the present algorithm predicts an initial blank shape in one
step for simplicity, as shown in Fig. 17, which is not horizontally symmetric, since the bottom of the oil pan is slanted. Neglecting complicated real forming processes, the prediction of blank shapes and strain distribution might show considerable deviation from the exact values. Nevertheless, the strain distribution obtained in Fig. 18 fully demonstrates and explains the location of severe deformation and the tendency of overall deformation with an almost-similar result to that obtained using incremental finite-element analysis. The thickness–strain distributions obtained along the lines, as shown in Fig. 18, are also not symmetric, since the bottom surface of the oil pan is inclined. Noting that the present algorithm provides information on an initial blank shape and strain distribution at the initial stage of the part-design procedure, the present algorithm can be considered as being indispensable for part and process design.
4. Conclusions An algorithm has been developed for the prediction of the initial blank shapes and strain distributions from desired final shapes in the sheet metal forming process. Analysis with the developed algorithm can help in the design of sheet metal parts and their manufacturing processes at the initial stage of product development and in order to reduce trial-and-error as well as cost. In this approach, the material is described by Hencky’s deformation theory and Hill’s anisotropic yield criterion. A finite-element formulation has been derived to minimize an approximated plastic potential energy, which considers various process variables such as the friction force, the blank-holding force and the drawbead force. The numerical and experimental results of the cylindrical and square cup drawing demonstrate that the present algorithm calculates initial blank shapes and strain distributions to within a small range of error. It is shown that the present algorithm can analyze complicated problems such as oil-pan drawing, by providing reasonable blank shapes and thickness-strain distributions. Although the present approach does not describe the deformation process precisely and follow its path, it provides useful information on process variables at the initial design stage with a rapid calculation.
References [1] T. Jimma, Deep drawing convex polygon shell researches on the deep drawing of sheet metal by the slip line theory, 1st report, Japan Soc. Tech. Plast. 11 (1970) 653. [2] V.V. Hazek, K. Lange, Use of slip line field method in deep drawing of large irregular shaped components, Proc. 7th NAMRC, 1979, p. 65.
C.H. Lee, H. Huh / Journal of Materials Processing Technology 82 (1998) 145–155 [3] M. Karima, Blank development and tooling design drawn parts using a modified slip line field based approach, ASME Trans. J. Eng. Ind. 111 (1989) 345. [4] J.H. Vogel, D. Lee, An analysis method for deep drawing process design, Int. J. Mech. Sci. 32 (1990) 891. [5] X. Chen, R. Sowerby, The development of ideal blank shapes by the method of plane stress characteristics, Int. J. Mech. Sci. 34 (1992) 159. [6] R. Sowerby, J.L. Duncan, E. Chu, The modeling of sheet metal stamping, Int. J. Mech. Sci. 28 (1986) 415. [7] G.N. Blount, P.R. Stevens, Blank shape analysis for heavy gauge metal forming, J. Mater. Process. Technol. 24 (1990) 65. [8] S.A. Majlessi, D. Lee, Further development of sheet metal forming analysis method, ASME Trans. J. Eng. Ind. 109 (1987) 330. [9] S.A. Majlessi, D. Lee, Development of multistage sheet metal forming analysis method, J. Mater. Shaping Technol. 6 (1988) 41. [10] S.A. Majlessi, D. Lee, Deep drawing of square-shaped sheet metal parts, part 1: finite element analysis, ASME Trans. J. Eng. Ind. 115 (1993) 102. [11] S. Levy, C.F. Shinh, J.P.D. Wilkinson, P. Stine, R.C. McWilson, Analysis of sheet metal forming to axisymmetric shapes, in: B.A. Niemeier, A.K. Schmeider, J.R. Newby (Eds.), Formability Topics — Metallic Materials, ASTM, Toronto, Canada, 1978, p. 238. [12] J.L. Batoz, Y.Q. Guo, P. Duroux, J.M. Detraux, An efficient algorithm to estimate the large strains in deep drwing, NUMIFORM ‘89, Fort Collins, CO, USA, A.A. Balkema, Rotterdam, 1989, p. 383. [13] J.L. Batoz, Y.Q. Guo, J.M. Detraux, An inverse finite element procedure to estimate the large plastic strain in sheet metal forming, Proc. 3rd Int. Conf. on Technology of Plasticity 3, 1990, Kyoto, Japan, p. 1403. [14] Y.Q. Guo, J.L. Batoz, J.M. Detraux, P. Duroux, Finite element procedures for strain estimations of sheet metal forming parts, Int. J. Numer. Methods Eng. 30 (1990) 1385. [15] Y.Q. Guo, J.L. Batoz, M.El. Mouatassim, J.M. Detraux, On the estimation of thickness strain in thin car panels by the inverse
.
[16]
[17] [18]
[19] [20] [21]
[22]
[23] [24]
[25]
[26]
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approach, in: J.L. Chenot, R.D. Wood, O.C. Zienkiewicz (Eds.), NUMIFORM ’92, Valbonne, France, A.A. Balkema, Rotterdam, 1992, p. 473. K. Chung, O. Richmond, Ideal forming — I. Homogeneous deformation with minimum plastic work, Int. J. Mech. Sci. 34 (1992) 575. K. Chung, O. Richmond, Ideal forming — II. Sheet forming with optimum deformation, Int. J. Mech. Sci. 34 (1992) 617. K. Chung, O. Richmond, Sheet forming process design based on ideal forming theory, in: J.L. Chenot, R.D. Wood, O.C. Zienkiewicz (Eds.), NUMIFORM ’92, Valbonne, France, A.A. Balkema, Rotterdam, 1992, p. 455. K. Chung, O. Richmond, The mechanics of ideal forming, J. Appl. Mech. 61 (1994) 176. R. Hill, A general theory of uniqueness and stability in elastoplastic solids, J. Mech. Phys. Solids 6 (1958) 236. N.M. Wang, A rigid – plastic rate-sensitive finite element method for modeling sheet metal forming processes, in: J.F.T. Pittman, et al. (Eds.), Numerical Analysis of Forming Processes, Wiley, New York, 1984, p. 117. H. Huh, S.S. Han, D.Y. Yang, Modified membrane finite element formulation considering bending effects in sheet metal forming analysis, Int. J. Mech. Sci. 36 (1994) 659. S.Y. Chung, H.W. Swift, A theory of tube-sinking, J. Iron Steel Inst. 170 (1952) 29. K.V. von Scho¨ning, R. Su¨nkel, M. Hillman, K.W. Blu¨mel, A. Wu¨lfing, Mathematical modelling bridges the gap between material and tooling, in: A. Makinouchi, E. Nakamachi, E. Onate, R.H. Wagoner (Eds.), NUMISHEET ’93, Isehara, Japan, 1993, p. 321. S.D. Liu, M. Karima, A one step finite element approach for production design of sheet metal stampings, in: J.L. Chenot, R.D. Wood, O.C. Zienkiewicz (Eds.), NUMIFORM ’92, Valbonne, France, A.A. Balkema, Rotterdam, 1992, p. 497. D.Y. Yang, Y.J. Kim, A Rigid – plastic finite element formulation for the analysis of general deformation of planar anisotropic sheet metals and its applications, Int. J. Mech. Sci. 28 (1986) 28.
Blank design and strain estimates for sheet metal forming processes by a finite element inverse approach with initial guess of linear deformation C.H. Lee, H. Huh * Department of Mechanical Engineering, Korea Ad6anced Institute of Science and Technology, 373 -1 Kusongdong, Yusonggu, Taejon 305 -701, South Korea Received 20 March 1997
Abstract A new finite element approach is introduced for direct prediction of blank shapes and strain distributions from desired final shapes in sheet metal forming. The approach deals with the geometric compatibility of finite elements, plastic deformation theory, minimization of plastic work with constraints and a proper initial guess. The algorithm developed is applied to cylindrical cup drawing, square cup drawing and oil pan drawing in order to confirm its validity by demonstrating reasonably accurate numerical results for each problem. Rapid calculation with this algorithm enables easy determination of various process variables for design of a sheet metal forming process. © 1998 Elsevier Science S.A. All rights reserved. Keywords: Blank design; Finite element inverse approach; Linear deformation
1. Introduction Sheet metal forming processes experiences very complicated deformation effected by process parameters such as the die geometry, the blank shape, the sheet thickness, the blank holding force, friction, lubrication and so on. Although these process parameters have an influence on the deformation mechanism and the quality of deformed parts, the optimum condition for process parameters is determined by intuition and experience, with trial-and-error. More systematic methods for the determination of the optimum conditions have been developed with the aid of computers and numerical analysis. Numerical analysis including finiteelement methods, can simulate complicated sheet metal parts and affords useful information on forming processes, thus reducing trial-and-error. However, numerical simulation is generally carried out with given process parameters and so requires numerical trial-anderror with enormous time and cost, to determine the optimum process parameters. For this reason, there * Corresponding author. Fax: +82 42 8693210/8695210. 0924-0136/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved. PII S0924-0136(98)00034-X
arises the necessity for some approaches to find one, or some, of the optimum process parameters directly. One of the important process parameters is the blank shape, which has a direct relationship with the quality of the deformed parts. Design methods for the blank shape have been widely studied by many researchers. Jimma [1], Hazek and Lange [2] and Karima [3] made use of the slip-line method to design the initial blank shape. Vogel and Lee [4] and Chen and Sowerby [5] used the characteristic of plane stress, while Duncan et al. [6] and Blount and Stevens [7] used geometric mapping to design the initial blank shape. These methods provide good guidance to design the initial blank shape, even though they neglect the height of the deformed parts, have geometric restrictions, or do not consider the deformation behavior of the materials. On the other hand, there have been several attempts to design the blank shape and estimate the distribution of the strain in a deformed part with deformation theory. Majlessi and Lee [8–10] extended the theory of Levy et al. [11] and applied it to axisymmetric problems and axisymmetric multi-stage problems, obtaining good results. Guo and Batoz [12–15] derived a formulation for field problems as an inverse method to obtain the initial
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tion theory and Hill’s anisotropic yield criterion. The calculation and minimization adopts an initial guess using inverse mapping with linear deformation, which is of great importance for good convergence. Minimization provides the optimum shape of the initial blank and the strain distribution in a deformed part under given friction conditions and blank-holding forces. The preapproximated inverse method enables the determination of process parameters within a small range of error with low computing time, affording the determination of the strain distribution in a deformed part before the process design.
2. Formulation Fig. 1. Kinematics of linear triangular membrane elements between the initial state and the final state.
blank shape and the thickness distribution in a deformed part. Chung and Richmond [16 – 19] suggested ideal forming with optimum deformation to design the initial blank shape and the intermediate deformed shapes. In this paper, the strain tensor at the deformed state is calculated from the initial state as a function of the coordinates by approximating the deformed shape to a system of discretized triangular membrane elements. The plastic work is then calculated element-wise and minimized satisfying the given constraints with the conjugate gradient method and the Newton – Raphson method. The plastic work is assumed to obey Hencky’s deforma-
Predicting an initial blank shape from a final deformed shape in a one-step calculation, the finite element approach using deformation theory has some different features from the conventional finite element method using the incremental theory. The problem reduces to minimizing the plastic potential energy, relating the initial state to the final state. In the problem, the given variables are the geometry of the final state and the thickness of the initial state, while the unknowns are the coordinates of the initial state and the thickness of the final state. These unknowns are determined by minimization of the objective function, which is an approximated plastic potential energy derived from the plastic work and the external work as a function of the unknowns.
Fig. 2. Transformation of elements from the final state to the initial state.
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147
Fig. 3. Deformed shapes of cylindrical cups with finite-element mesh geometry obtained from CAD models: (a) cup height =20 mm; (b) cup height =30 mm; and (c) cup height=40 mm.
2.1. Plastic potential
thoroughly by various numerical examples and corresponding experiments.
A deformable body in equilibrium satisfies the principle of virtual work as the necessary and sufficient condition for the stress field. Hill assumed the existance of a plastic potential that has its minimum value when a plastic body is in equilibrium [20]: min. C =
&
D
s:o; dV −
&
(D
t · 7 dG
(1)
The above equation holds in any bodies in the plastic state with large deformation, regardless of their constitutive relationships. In this formulation, the plastic potential energy in the final state is derived as a function of the initial coordinates, neglecting the intermediate deformation path, which might lead to absurd results with remarkable errors. This kind of analysis therefore, must be modulated carefully and examined
2.2. Kinematics The geometry of the initial and final state is discretized by linear triangular membrane elements. Fig. 1 shows the schematic elements of the initial and final states. By the assumption of proportional loading and the minimum plastic work path [19,26], the strain can be identified as the logarithmic strain which ensures the incompressibility of the material during large deformation. The logarithmic strain components oi (X) are expressed as Eq. (2) from the Green deformation tensor [21]:
Æo xx Ç Æln l 1 cos2 u+ ln l2 sin2 u Ç [oi (X)] = à o yy à = Ãln l 1 sin2 u+ ln l2 cos2 u à Èo xy É È(ln l 1 − ln l2) sin u cos u É (2) 6where l1, l2 are defined by the principal values of the Green deformation tensor and u designates the direction of l1 with respect to the X coordinates.
Fig. 4. Finite-element meshes for an initial guess and the computed blank shape of a cylindrical cup: (a) initially-guessed shape by linear inverse mapping; and (b) computed blank shape (cup height = 30 mm).
Fig. 5. Computed blank shapes for various cup heights in cylindrical cup drawing.
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148
Fig. 6. A photograph of cylindrical cups after deep drawing: (a) cup height =20 mm; (b) cup height=30 mm; and (c) cup height= 40 mm.
' '
n n
Æ (C + C ) (C1 −C2) 2 1 2 + +C 23 Ã 2 2 Æl 1 Ç Ã (C 1 + C2) (C1 −C2) 2 − Ãl 2 Ã = Ã +C 23 2 2 Èu É Ã l 1 −C1 Ã tan − 1 C3 È
Ç Ã 1/2Ã Ã Ã Ã É
1/2
(3) The Green deformation tensor can be constructed by two pairs of vectors in Fig. 1 as follows:
The logarithmic strain components in Eq. (1) are constant in a linear triangular element. Since the coordinates of the nodal points in the final state, x, are known, the strain components become functions of the coordinates of the nodal points at the initial state, X. The coordinates X are updated at every iteration of the minimization process.
2.3. Constituti6e relationship The stress–strain rate relationship, which is derived from the associated flow rule and Hill’s anisotropic
[C11 C22 C12]T [C1 C2 C3]T 12G3y +13G2y − (12 + 13 − 11)G2y G3y Æ Ç 12G3x 2 +13G2x 2 − (12 + 13 − 11)G2x G3x à à 1 = à à (G3yG2x −G3xG2y )2 1 + 13 − 11 à −12G3x G3y −13G2x G2y + 2 (G2x G2y + G2y G3x )à 2 È É 2
2
where 1i =g 2ix +g 2iy +g 2iz.
(4)
yield criterion for plane stress, is integrated from the initial state to the final state. Then, the integrated relation is expressed as Eq. (5) in an inverted form. The concept of Eq. (5) is from the Hencky’s deformation theory, i.e. the strain components in the final state are assumed to be proportional to the stress components in the final state. Although the deformation theory should be restricted to infinitesimal deformation, it is assumed that the deformation follows the proportional loading path, allowing errors from the assumption of proportional loading:
Æs xx Ç ÆAB A 0 ÇÆo xx Ç 2 s¯ à s yy à = à A AC 0 Ãà o yy à Ès xy É 3 o¯ È 0 0 ADÉÈo xy É
(5)
where
Fig. 7. Comparison of the flange contours in the final shape between those of the desired CAD models and the experimental contours.
A=
r(2+ r) , (1+ 2r)
D=
1 r
B=
(1+ r) , r
C=
(1+r) , r (6)
C.H. Lee, H. Huh / Journal of Materials Processing Technology 82 (1998) 145–155
149
Fig. 8. Deformed shapes of square cups with finite-element mesh geometry obtained from CAD models: (a) cup height = 20 mm; (b) cup height =30 mm; and (c) cup height=40 mm.
The effective strain corresponding to Eq. (5) can be expressed as: o¯ =
'
2 A[Bo 2xx + 2oxxoyy +Co 2yy +2Do 2xy] 3
(7)
E
Wp(X) = % e=1
& & De
ef 0
2 s¯ D o do t dA 3 o¯ ab a b
(9)
Eq. (9) shows the plastic work, Wp(X), is also a function of the nodal coordinates, X, at the initial state.
where r in Eq. (6) is the Lankford value.
2.5. Boundary conditions
2.4. Plastic deformation energy
Sheet metal has a relatively large surface area compared to its volume. While the boundary conditions applied to a sheet surface change continuously during deformation, the deformation theory used in the present analysis retains a rigorous modeling of the boundary conditions. For this reason, a simplified method which considers the friction force, the blank-holding force and the draw-bead force, is proposed as follows.
When the domain of the final state is discretized by finite elements, the sum of the plastic deformation energy for each element can be calculated by use of Eq. (8): E
Wp(X) = % e=1
& & De
ef
s:do dV
(8)
0
where the stress and strain components are defined by Eqs. (2) and (5), respectively and of, is the strain tensor of the final state. Since a linear triangular element is adopted, the thickness, as well as the stress and strain components, are constant within an element. Thus, Eq. (8) is simply rewritten as Eq. (9):
2.5.1. Friction boundary conditions The friction force induced by the normal force is considered for the punch and die interface surface. The normal force Fn at a node can be obtained from the equilibrium of sheet metal:
Fig. 9. Finite-element meshes for an initial guess and the computed blank shape of a square cup: (a) initially-guessed shape by linear inverse mapping; and (b) computed blank shape (cup height = 30 mm).
Fig. 10. Four shapes of finite element meshes for an initial guess: (a) by projection; (b) by linear mapping with a scale factor of 0.75; (c) by linear mapping with a circular contour; and (d) by linear mapping with a scale factor of 1.15 (cup height= 30 mm).
C.H. Lee, H. Huh / Journal of Materials Processing Technology 82 (1998) 145–155
150
angle at the node for each element and the reaction force at the node for each element; nf is the outward unit normal vector at the final state. The normal force in Eq. (10) is an averaged value at the nodal point and supposed to exist where the sheet metal is in bending. The frictional work is then written as follows:
& & n uf
E
Wf(X) = %
e=1
(De
tf du dG
0
= − % mFn tanh a node
U U U
U U
(12)
where tf, m and U are the frictional stress, the friction coefficient and the tangential displacement; uf is the total displacement from the initial state to the final state and U is obtained from Eq. (13): U(X) =[(x − X) −(ni · (x− X))ni]
Fig. 11. Computed blank shapes from four initial guesses: (a) by projection; (b) by linear mapping with a scale factor of 0.75; (c) by linear mapping with circular contour; and (d) by linear mapping with a scale factor of 1.15 (cup height= 30 mm).
(13)
The hyperbolic tangent function in Eq. (12) stabilizes the numerical oscillation when a node in contact becomes almost stuck [22]. The coefficient a is a control number for good convergence, ranging from 1 to approximately 10. ni is the unit normal vector at the initial state.
m
Fn =
*
% uk F k k=1 m
% uk F k
k=1
where Fk =
&
*
(De
· nf
t dG =
(10)
&
s: De
(o dV (u
(11)
In the above equation, m, uk and Fk are the total number of elements that contain a designated node, the
2.5.2. Blank-holding force and draw-bead force The blank-holding force prevents the sheet metal from wrinkling and controls the drawing of sheet metal from the flange. According to Chung and Swift’s experiment [23], most of the blank-holding force is concentrated at the outer edge of the flange where the sheet metal thickens. Thus, the frictional work by the blankholding force can be evaluated at the nodes in the outer edge of a blank sheet:
& & n uf
Eb
Wb(X) = %
e=1
(De
tb du dG
0
= − % mFb tanh a node
U U U
U U
(14)
where Fb is the blank-holding force converted into the nodal values at the outer edge. The work performed by the draw-bead force can be evaluated with the same concept as for the blank holding force:
& & n uf
Ed
Wd(X) = %
e=1
(De
td du dG
0
= − % mFd tanh a node
Fig. 12. Computed blank shapes for various cup heights in square cup drawing.
U U U
U U
(15)
where the draw-bead force Fd can be obtained from either experiment or numerical simulation. The blankholding force, the draw-bead force and the location of the draw-bead can be regarded as process variables determined at the design stage by the present simulation algorithm.
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151
Fig. 13. A photograph of square cups after deep drawing: (a) cup height =20 mm; (b) cup height= 30 mm; and (c) cup height=40 mm.
2.6. Minimization of the objecti6e function
imated plastic potential energy, as follows:
An objective function derived from the plastic potential energy can be minimized to solve inverse problems in sheet metal forming processes. However, the problems do not provide the prescribed static and kinematic boundary conditions. Without incremental steps which follow deformation paths, the punch force and the intermediate deformed shape cannot be calculated and the plastic potential energy in any stage is almost impossible to obtain. In order to overcome these difficulties, an approximated plastic potential energy is defined by the difference between the plastic deformation energy and the equivalent external work performed. The equivalent external work can be obtained from the force equilibrium equation: Fp + Ff +Fb + Fd = 0
or
Fp = −Ff −Fb −Fd
(16)
where Fp is the punch force, Ff is the friction force due to bending, Fb is the blank-holding force and Fd is the draw-bead force. The concept of force equilibrium and the principle of virtual work enables calculation of the so-called equivalent external work and thus, an approx-
min. C(X) Wp(X) − Wf(X) − Wb(X) − Wd(X)
where, Wp(X) is the plastic deformation energy; Wf(X) is the work done by friction due to bending; Wb(X) is the work done by the blank-holding force; and Wd(X) is the work done by the draw-bead force. The formulation for the minimum of C(X) is derived to construct a matrix equation for the conjugate gradient method or the Newton–Raphson method. The local minimum of C(X) can be satisfied when the first derivative of C(X) has a stationary value. The first derivative is expressed as follows: min. C(X)UR(X)
(C(X) =0 (X
(18)
where
&
R(X) E
= % e=1
− %
s:
&
De
Eb
e=1
Fig. 14. Comparison of the flange contours in the final shape between those of the desired CAD models and the experimental contours.
(17)
E (o dV − % (X e=1
(De
&
tb
&
(De
(u dG − % (X e=1 Ed
tf
(u dG (X
(De
td
(u dG (X
(19)
Fig. 15. Comparison of the thickness – strain distribution between computed results and experimental results for square cups (cup height=30 mm).
C.H. Lee, H. Huh / Journal of Materials Processing Technology 82 (1998) 145–155
152
Fig. 16. Deformed shape of the oil pan with the finite-element mesh geometry obtained from the CAD model.
for the effective stress – strain relationship s¯ = k(o0 + o¯ )n. The conjugate gradient method requires only the first derivatives in Eq. (19). However, the Newton– Raphson method requires the second derivatives and is expressed as follows:
(R(X) (X
n
{dX} = − {R(X)}n
(20)
(n)
X(n + 1) =X(n) +b dX where E (R(X) = % (X e=1
& & De
E
Eb
− % e=1
&
− %
e=1
(21)
(s (o ( 2o : + s: dV (X (X (X2
(tf (u dG (X (X (De
Ed (tb (u dG − % e=1 (De (X (X
&
(td (u dG (De (X (X
Fig. 18. Thickness – strain distributions of the oil pan across line A1 – A2 and line B1 – B2.
(22)
where b is the deceleration coefficient ranging from 0 to 1. The first and second derivatives of the strain components in Eqs. (19) and (22) are derived as:
3 3 4 (ep (ep (la (Cb (Gg = % % % (Xk a = 1 b = 1 g = 1 (la (Cb (Gg (Xk
(p=1− 3; ( 2op ( = (Xj(Xk (Xj
k= 1−6) (ep lla (Cb (Gg a = 1 b = 1 g = 1 (la (Cb (Gg (Xk 3
3
%
4
% %
n
( 2op (la (Cb (Gg r = 1 s = 1 t = 1 a = 1 b = 1 g = 1 (lr (la (Cb (Gg (Xk 3
3
4
3
3
= % % % % × × ×
4
% %
(23)
3 3 4 3 4 (lr (Cs (Gt (op ( 2la + % % % % % (Cs (Gt (Xj a = 1 r = 1 s = 1 b = 1 g = 1 (la (Cr (Cb
(Cb (Gr (Gr (Xk
3 3 4 4 (Cr (Gs + % % % % (Gs (Xj a=1 b=1 r=1 g=1
(op (la ( 2Cb (Gg (Gr (la (Cb (Gr (Gg (Xk (Xj (oi (la (Cb ( 2Gg a = 1 b = 1 g = 1 (la (Cb (Gg (Xk (Xj 3
+ % Fig. 17. Computed blank shape for the oil pan from the CAD model.
3
4
% %
(p=1 −3;
j,k =1− 6)
(24)
C.H. Lee, H. Huh / Journal of Materials Processing Technology 82 (1998) 145–155
The conjugate gradient method and the Newton– Raphson iteration start with an initial guess of a point set X0 obtained in the next section.
2.7. Initial guess by in6erse mapping with linear deformation
E
% e=1
&
E
[B]T[D][B] dV{U} = % Ve
e=1
&
[B]T[D][o0] dV Ve
to the drawing of an oil pan, which has more complicated geometries. The material properties and the process variables are shown below for simulation and experiment into cylindrical and square cup drawing. Stress–strain curve
The non-linearity that comes from the large deformation and the material properties makes it difficult to solve Eq. (17), which essentially needs an initial guess that determines appropriate points X0 before solving with the conjugate gradient method or the Newton– Raphson method. The idea of the initial guess used in the present analysis is a linear mapping of the deformed surface of the final state to a flat surface of the initial state. The mapping procedure discretizes the surface into triangular finite elements and assumes that the material in each element deforms linearly. In order to obtain approximate mapping from the deformed surface to an initial flat surface, two types of transformation between the final and initial state are introduced (Fig. 2). The scheme for an initial guess is as follows. Step 1. Carry out two types of transformation for each element from the final state to the initial state, as shown in Fig. 1. Calculate initial strain o0 between the two transformed elements and assume that an initial strain yields the internal stress. Step 2. Solve the elastic recovery using an elastic finite-element method expressed in Eq. (25) for the elements of‘ the second transformation:
(25)
Step 3. If the norm of the right-hand side of Eq. (25) is less than unity, go to Step 4. Otherwise, update the coordinates and go back to Step 2. Step 4. Modulate the area of the calculated shape as the same area of the final state. This modulation is performed because the initial guess scheme is a reverse process to the real deformation process. The modulated shape is used as an initial guess of a point set X0.
3. Numerical results The algorithm described above has been implemented in a finite-element code and applied to several sheet metal forming examples for validation. The blank shapes and thickness distributions of a cylindrical and a square cup have been obtained as a bench-mark test. In order to check the validity of the present algorithm, experiments have been carried out for cylindrical and square cup drawing with blank specimens prepared for the calculated blank shape. As a demonstration of its versatility, the present algorithm has also been applied
153
s¯ = 54.5 (0.00436+o¯ )0.263 (kgf mm−2) r= 1.87 t= 0.8 mm
Lankford value Initial sheet thickness Friction coefficient m= 0.15 Blank-holding force Fb = 4000 kgf
3.1. Deep drawing of a cylindrical cup The present algorithm is first applied to the deep drawing of a cylindrical cup for demonstration of its validity. The geometries of cylindrical cups discretized with finite elements are shown in Fig. 3. The height of the cylindrical cups are 20, 30 and 40 mm, respectively. The diameter of the cups is 60 mm and the width of the flange is 5 mm. For the sake of symmetry, only one quarter of the cup is considered in the calculation. Fig. 4(a) is an initial guess obtained from inverse mapping with linear deformation and Fig. 4(b) is a blank shape calculated in the analysis from Fig. 4(a) as an initial guess. Both figures show roundness on the outside rim, which demonstrates the stable-mesh independency of the present algorithm. The initial blank shapes for the three cup heights are depicted in Fig. 5, showing that the size of the initial blank increases with the cup height. Blank shapes for normal anisotropic sheet metal are calculated as near-circles. Deep-drawing experiments were performed with the specimens of the calculated blank shapes in Fig. 5, with Fig. 6 presenting a photograph of the cylindrical cups formed in the experiments. The flanges of the deepdrawn cups in the experiments are compared with the final shapes, in the analysis shown in Fig. 7. The difference in Fig. 7 can be explained both by simulation error and experimental error. The simulation error is mainly due to the use of deformation theory, neglecting the anisotropy of the sheet metal and the boundary conditions with simplified equations, while the experimental error is mainly due to the eccentricity of the specimen placement and discrepancy in the distribution of the blank-holding forces. Although the present method calculates as a one-step procedure and has many problems, the error is found to be relatively small, demonstrating that the present method can predict a blank shape within a small range of error.
3.2. Deep drawing of a square cup Square cups have been considered in order to obtain their blank shapes and thickness strain distributions.
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The geometries of the square cups are discretized with finite elements, as shown in Fig. 8. Fig. 9(a) is an initial guess obtained from inverse mapping with linear deformation and Fig. 9(b) is a blank shape calculated in the analysis from Fig. 9(a) as an initial guess. In order to demonstrate the independency of an initial guess, four arbitrary initial guesses, as shown in Fig. 10, are used to calculate the initial blank shape. Fig. 11 fully demonstrates the independency of an initial guess with the present algorithm, since the initial blank shapes obtained are almost the same, although the initial guesses are very different from each other. The initial blank shapes for three cup heights are depicted in Fig. 12. Experiments have been carried out with the calculated blank shapes in Fig. 12, the photographs of the square cups formed in the experiment being shown in Fig. 13. The flange shapes obtained from deep drawing of the initial blanks in experiments are compared with those of CAD models in Fig. 8. The comparison presented in Fig. 14 shows that there are differences in the flange shape between theat desired and that obtained by experiment using calculated blanks, there being an excess in the 45° direction and deficiency in the rolling direction. This is due to the fact that the present algorithm does not take account of the concentration of the blank-holding force in the region where sheet metal becomes thicker during deformation. The thickness-strain distributions along the transverse and diagonal direction of a square cup are shown in Fig. 15. The thickness-strain distributions predicted by the present method are in moderate agreement with the experimental results in the upper region and the flange region of a cup, except for the region near to the wall. Since the incremental finite-element method also calculates the different thickness-strain distribution from experimental results [22], this result obtained with one-step calculation could be acceptable.
3.3. Deep drawing of an oil pan The present algorithm was applied to a more complicated problem of oil-pan drawing for demonstration of its capability and versatility. An oil pan for under an automobile engine is shown in Fig. 16, with finite-element descritization. The geometry of the oil pan is discretized by 1460 elements and 766 nodes. The oil pan is usually manufactured by two steps of drawing in a press shop and needs careful process design. This process has been simulated with incremental finite-element analysis by several researchers [24]. Liu and Karima [25] calculated the blank shape for an oil pan by applying infinitesimal strain equation to a large deformation problem and solving the problem in one step. Regardless of how complicated the process is, the present algorithm predicts an initial blank shape in one
step for simplicity, as shown in Fig. 17, which is not horizontally symmetric, since the bottom of the oil pan is slanted. Neglecting complicated real forming processes, the prediction of blank shapes and strain distribution might show considerable deviation from the exact values. Nevertheless, the strain distribution obtained in Fig. 18 fully demonstrates and explains the location of severe deformation and the tendency of overall deformation with an almost-similar result to that obtained using incremental finite-element analysis. The thickness–strain distributions obtained along the lines, as shown in Fig. 18, are also not symmetric, since the bottom surface of the oil pan is inclined. Noting that the present algorithm provides information on an initial blank shape and strain distribution at the initial stage of the part-design procedure, the present algorithm can be considered as being indispensable for part and process design.
4. Conclusions An algorithm has been developed for the prediction of the initial blank shapes and strain distributions from desired final shapes in the sheet metal forming process. Analysis with the developed algorithm can help in the design of sheet metal parts and their manufacturing processes at the initial stage of product development and in order to reduce trial-and-error as well as cost. In this approach, the material is described by Hencky’s deformation theory and Hill’s anisotropic yield criterion. A finite-element formulation has been derived to minimize an approximated plastic potential energy, which considers various process variables such as the friction force, the blank-holding force and the drawbead force. The numerical and experimental results of the cylindrical and square cup drawing demonstrate that the present algorithm calculates initial blank shapes and strain distributions to within a small range of error. It is shown that the present algorithm can analyze complicated problems such as oil-pan drawing, by providing reasonable blank shapes and thickness-strain distributions. Although the present approach does not describe the deformation process precisely and follow its path, it provides useful information on process variables at the initial design stage with a rapid calculation.
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