ELSEVIER Journal ofMaterials Processing Technology 63 (1997) 645-650

JoumaJof

MaterialsProcessingTechnology

Blank Design and Strain Prediction of Automobile Stamping Parts by an Inverse Finite ElementApproach

C.H. Lee and H. HuhDept. ofMechanical Engineering, Korea Advanced Institute ofScience and Technology,

373-1 Kusongdong, Yusonggu, Taejon 305-701, Korea

Abstract

A new finite element approach is introduced for direct prediction of blank shapes and strain distributions from desired final shapesin 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 automobile stampingparts such as an oil pan and a front fender in order to confirm its versatility of application by demonstrating the reasonable numericalresults in stamping processes. Rapid calculation with this algorithm enables easy determination of various process variables for designof sheet metal forming process.

1. Introduction

Sheet metal forming processes experiences very complicateddeformation effected by process parameters such as the diegeometry, the blank shape, the sheet thickness, the blankholding force, friction, lubrication, and so on. These processparameters have to be determined for the optimum formingcondition before the process design. The determination yet relieson the experience and intuition or the time-consuming computeranalysis such as incremental finite element methods for smallmodification after the process design.

The determination of blank shapes has been attempted invarious methods such as the slip-line method[I-3], the character­istic of plane stress[4-5], and the geometric mapping[6-7]. Thesethree methods provide good guidance to determine the initialblank shape even with some restrictions. More versatile methodscould be to design blank shape with the deformation theory. Guoand Batoz proposed an inverse method to obtain the initial blankshape and the thickness strain distribution in a deformed part[8­9]. Chung and Richmond suggested ideal forming with theoptimum deformation to design the initial blank shape andintermediate deformed shapes[ 10,11].In this paper, an inverse finite element approach is introducedfor direct prediction of blank shapes and strain distributions inthe desired final shape. In the formulation, the strain tensor atthe deformed state is calculated from the initial state as afunction of the coordinates. The plastic work is then calculatedelement-wise to construct the objective function defined bydifference between the internal plastic work and the externalwork. The possible external work is calculated from thefrictional force, the blank holding force, and the draw-bead forceinstead of the punch force that is unable to calculate easily. In

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the calculation, the material is assumed to obey Hencky'sdeformation theory and Hill's anisotropic yield criterion. Theobjective function calculated is minimized with the conjugategradient method and the Newton-Raphson method from theinitial guess using inverse mapping with linear deformation. Theminimization provides the optimum shape of an initial blank andthe strain distribution in a deformed part under the given frictionconditions and the blank holding forces.

The present algorithm is successfully applied to complicatedautomobile stamping parts such as front fenders and oil pans.The results provide not only the optimum shape of an initialblank but the thickness distribution in a deformed part toconfirm the possibility of production. The calculation is also ableto consider the proper friction conditions and blank holdingforces. The pre-approximated inverse method enables thedetermination of process parameters including the optimuminitial blank shape, the blank holding force, and the draw-beadforce with small range of error in good r.;omputing effort.

2. Formulation

Predicting an initial blank shape from a final deformed shapein one step calculation, the finite element approach using thedeformation theory have some different features from theconventional finite element method using the incremental theory.The problem reduces to minimize the plastic potential energy,relating the initial state to the final state. In the problem, thegiven variables are the geometry of the final state and thethickness of the initial state, while the unknowns are thecoordinates of the initial state and the thickness of the final state.These unknowns are determined by minimization of theobjective function which is an approximated plastic potential

646 CH. Lee. H. Huh / Journal of Materials Processing Technology 63 (1997) 645-650

energy derived from the plastic work and the external work as afunction of the unknowns.

the final state.

2.1. Plastic deformation energy (2a)

KinematicsThe geometry of the initial and final state is discretized by

linear triangular membrane elements. Fig.l shows schematicelements of the initial and final states.

(2b)

And the effective strain corresponding to the Eqn (2) can beexpressed as

where rx ' r45 and ry in Eqn (2) and (3) are the Lankford values

for the orientations of 00 ,45 0 and 90° with respect to the rollingdirection.

(3)

(2c)

(1)

The logarithmic strain components tj (X) are expressed as Eqn(I) from the Green deformation tensor. AI' A2 are defined by

the principal values of the Green deformation tensor and the edesignates the direction of the AI with respect to the X

coordinates. The Green deformation tensor can be constructed bytwo pairs of vectors in Fig.1 as below. These logarithmic straincomponents in Eqn (I) are constant in a linear triangularelement. Since the coordinates of nodal points in the final state,x, is known, the strain components become functions of thecoordinates of nodal points at the initial state, X. Thecoordinates X are updated at every iteration ~f the minimizationprocess.

GZ

Plastic deformation energyWhen the domain of the final state is discretized by finite

elements, the sum of the plastic deformation energy for eachelement can be calculated by Eqn (4).

where the stress and strain components are defined by Eqn (I)

and (2) respectively. Since a linear triangular element isadopted, the thickness as well as the stress, strain componentsare constant within an element. Thus, Eqn (4) is simply rewrit­ten as Eqn (5)

(4)E

Wp(X) = L f<Jijtij dVe=! Y,

(x." Y3' Z3)

~(Xl' YI' Zl) g3 ,: (X2' ~z' Zz)

(X3, Y3)

z

~xFig.l Kinematics in a linear triangular membrane elements be­tween the initial state and the final state.

(5)

Eqn(5) shows the plastic work, Wp(X), is also the function of

the nodal coordinates X at the initial state.

Constitutive relationThe stress-strain rate relation, which is derived from the

associated flow rule and the Hill's anisotropic yield criterion forplane stress, is integrated from the initial state to the final state.Then, integrated relation is expressed as Eqn (2) in an invertedform. The concept of the Eqn (2) is from the Hencky'sdeformation theory. That is, the strain components in the finalstate are assumed to be proportional to the stress components in

2.2. Boundary conditions

Sheet metal has relatively large surface area compared to itsvolume. While the boundary conditions applied to the sheetsurface change continuously during the deformation, the defor­mation theory used in the present analysis restrains a rigorousmodeling of the boundary conditions. For the reason, asimplified method which considers the friction force, the blankholding force, and the draw-bead force is proposed as follows.

CH. Lee. H. Huh I Journal of Materials Processing Technology 63 (1997) 645-650 647

Friction boundary conditions

Friction force induced by the normal force is considered forthe punch and die interface surface. And the normal force Fa at anode can be obtained from the equilibrium of sheet metal.

be regarded as process variables determined at the design stageby the present simulation algorithm.

2.3. Objective function minimization

where Ft , /--l, V t is the tangential force, the friction coefficient

and the tangential displacement. V t is obtained from Eqn (8).

(11)or

where Fp is the punch force, Fb is the blank holding force, Fd isthe draw-bead force, and Fr is the friction force due to bending.The concept of the force equilibrium and the principle of virtualwork enables calculation of the so called equivalent externalwork and thus an approximated plastic potential energy asfollow:

An objective function derived from the plastic potentialenergy could be minimized to solve inverse problems in sheetmetal forming processes. The problem, however, does notprovide the prescribed static and kinematic boundary conditions.Without incremental steps which follow deformation paths, thepunch force and the intermediate deformed shape can not becalculated and the plastic potential energy in any stage is almostimpossible to obtain. In order to overcome the difficulties, anapproximated plastic potential energy is defined by thedifference between the plastic deformation energy and theequivalent external work done. The equivalent external workcould be obtained from the force equilibrium equation

(6), where

m

I8k Fk

lI~e,F,IINF =n

In the above equation, m, 8k ,Fk are the total number of

elements which contain a designated node, the angle at the nodefor each elements, and the reaction force at the node for eachelements. N is the outward unit normal vector at the nodal point.The normal force in Eqn (6) is an averaged value at the nodalpoint and supposed to exist where the sheet metal is in bending.The frictional work is then written as follow:

Wf(X) = IFt ·V t =- I [/--lFntanh(exlvd)ll~tll]' V t (7)node node t

(8)(12)

where the subscript i means the components and the superscript jmeans the j-th node.

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, Wd (X) is the work done by the

draw-bead force.Minimization of the objective function defined starts with an

initial guess of a point set Xo obtained in the next section. Theminimization procedure adopts the conjugate gradient method atfirst to stabilize the convergence and the Newton-Raphsonmethod for the subsequent iterations. Since the conjugategradient method is characterized by the stable but slowconvergence, the method is used before the Newton-Raphsonmethod and limited to a certain amount of iteration with thefollowing criterion. And then, Newton-Raphson method is usedfor the continuing minimization process.

The hyperbolic tangent function in Eqn (7) stabilizes thenumerical oscillation when a node in contact becomes almost

stuck. The coefficient ex is a control number for good conver­gence ranged from I to approximately 100.

Blank holding force and draw-bead forceThe blank holding force prevents sheet metal from wrinkling

and controls drawing of sheet metal from the flange. Accordingto Chung and Swift's experiment[12], most of the blank holdingforce concentrates at the outer edge of the flange where sheetmetal thickens. So, the frictional work by the blank holding forcecould be evaluated at the nodes in the outer edge of a blanksheet.

Wb(X) = IFt · V t =- I [/--lFt,tanh(ex1vtI) II~tll]' V t (9)node node t

where Fb is the blank holding force converted into the nodalvalues at the outer edge.

The work done by the draw-bead force can be evaluated withthe same concept as the blank holding force.

max[ ±(d'¥(~I)J]~8J i=l aXt

(13)

(10)

where the draw-bead force Fd could be obtained from either theexperiment or the numerical simulation. The blank holdingforce, the draw-bead force, and the location of the draw-bead can

2.4. Initial guess by inverse mapping with linear deformation

The nonlinearity which comes from the large deformation andthe material properties makes it difficult to solve Eqn (12). Eqn(12) essentially needs an initial guess which determinesappropriate points Xo before solving with the conjugate gradient

648 c.H. Lee, H. Huh / Journal of Materials Processing Technology 63 (1997) 645-650

method. The idea of the initial guess used in the present analysisis a linear mapping of the deformed surface of the final state to aflat surface of the initial state. The mapping procedurediscretizes the surface into triangular finite elements andassumes that the material in each element deforms linearly. Inorder to obtain approximate mapping from the deformed surfaceto an initial flat surface, intermediate surfaces between the finaland initial state are introduced. The scheme for an initial guessis as follows:Step I. Make the n number of intermediate surfaces including

the surface of the final state and project the first surfaceon the plane. Set k = O.

Step 2. Let k = k + I. Assume the k-th intermediate surfacedeform to the (k-I )-th flat surface and calculate the

strain £* caused by the deformation. Then the k-th flatsurface is obtained by solving an elastic finite element

problem considering the recovery of the strain £ * .Step 3. If k < n, go to Step 2. Otherwise, go to Step 4.Step 4. Modulate the area of the n-th flat surface to be less than

the area of the final state by 5 to 10%. This modulationis effective because the initial guess scheme is a reverseprocess to the real deformation process and sheet metalis generally elongated with deformation. After themodulation, the n-th flat surface is used an initial guessof a point set Xo .

The number of the intermediate surfaces is set to be approxi­mately between 3 and 20.

3. Numerical results

good agreement with results in literature[5], when the material issteel sheet for autobody stamping.

Fig.2 Deformed shape of square cups with finite element meshgeometry obtained from CAD models: cup height= 30mm.

The present algorithm described above has been imple­mented in a finite element code and applied to the several sheetmetal forming examples for its validation. Blank shapes andthickness distributions of a square cup have been obtained as abench mark test. In order to check the validity of the presentalgorithm, experiments have been carried out for square cupdrawing using blank specimens prepared for the calculated blankshape. As a demonstration of its versatility, the presentalgorithm has been also applied to drawing of an oil pan whichhas more complicated geometries.

Fig.3 Finite element meshes for an initial guess and a computedblank shape of a square cup: (a) Initial shape guessed by linearmapping; (b) Computed optimum blank shape.

0.2 -,---------------------,£:, £:, £:, Experiment, Transverse Direction-- Present Theory, Transverse Direction

0.15- 000 Experiment, Diagonal Direction- - - Present Theory, Diagonal Direction

0.1 -

3.1. Deep drawing ofa square cup

The material properties and the process variables forsimulation and experiments of square cup drawing are as below: 0.0

-0.1

I:'@.... 0.05­

+oJ[fJ

-015

[f)[f)<J)

I:~ -0.05

:.aE-<

cr = 54.5(0.00436+ £)0.263 (kgf / mm 2 )

rx = 1.92, r45 = 1.60,ry = 2.35

t = 0.8 (mm)11=0.15

Fb = 4000 (kgf)

Stress-strain curveLankford value

Initial sheet thicknessFriction coefficientBlank holding force

Square cups have been considered to obtain their blankshapes and thickness strain distributions. The geometry of squarecups are discretized with finite element as shown in Fig.2.Fig.3(a) is an initial guess obtained from inverse mapping withlinear deformation and Fig.3(b) is a blank shape calculated usingFig.3(a) as an initial guess. The result of the blank shape is in

-0.2 0 10 20 30 40 50 60 ~oInitial Distance from the Center (mm)

Fig.4 Comparison of the thickness strain distribution betweenthe computed result and the experimental one.

CH. Lee, H. Huh I Journal of Materials Processing Technology 63 (1997) 645-650 649

The thickness strain distributions along the line AI-A2 andthe line BI-B2 in Fig.6 are shown in Fig.7. From the results inFig.7, we can predict the location and amount of the maximumstrain.

0.3

0.2 -----A-- lin e Al - A2-----e-- line Bl - B2

>=10.1

'8r-.....,

1JJ 0.0[JJ[JJ

III>=1 -0.1~

.~

..c:E-o -0.2

-0.3

The thickness strain distributions along the transverse anddiagonal direction of a square cup are shown in FigA. Thethickness strain distributions predicted by the present method arein moderate agreement with the experimental results in theupper region and the flange region of a cup except the regionnear the wall. Since the incremental finite element method alsocalculates the different thickness strain distribution fromexperimental results, this result with one step calculation couldbe acceptable.

3.2. Deep drawing ofan oil pan

The present algorithm has been applied to a more compli­cated problem of oil pan drawing for demonstration of itscapability and versatility. An oil pan under an automobile engineis shown in Fig.5 with finite element descritization. Thegeometry of the oil pan is discretized by 1460 elements and 766nodes. The oil pan is usually manufactured by two step ofdrawing in a press shop and need a careful process design. Liuand Karima[13] calculated a blank shape for an oil pan byapplying infinitesimal strain equation to a large deformationproblem and solving the problem in one step.

Fig.5 Deformed shape of an oil pan with finite element meshgeometry obtained from a CAD model.

o 100 200 300 400 500

Distance from the Point Al and BI (mm)

Fig.7 Thickness strain distributions for an oil pan along the lineAI-A2 and the line BI-B2.

3.3. Front fender drawing

The complicated geometry of a front fender for a bench marktest in NUMISHEET[14] is used as the final example. Becauseof the complicated geometry, it takes tremendous time tosimulate the stamping process with an incremental finite elementmethod. The present method, however, can predict blank shapesand the strain distributions in short time without difficulty. Thegeometry discretized by 2386 elements and 1245 nodes is shownin Fig.8 and the blank shape of the front fender predicted by thepresent algorithm is shown in Fig.9. It took 32 minutes in CPUtimes with HP715175 to solve the problem.

Fig.6 Computed optimum blank shape for an oil pan from a CADmodel.

Fig.8 Deformed shape of a front fender with finite element meshgeometry obtained from a CAD model.

650 CH. Lee, H. Huh / Journal of Materials Processing Technology 63 (1997) 645-650

Fig.9 Computed optimum blank shape for a front fender from aCAD model.

No matter how the process is complicated, the presentalgorithm predicts an initial blank shape in one step forsimplicity as shown in Fig.9. Neglecting the complicated realforming process, predicted blank shapes and strain distributionmight show considerable deviation from the exact values.Nevertheless, the strain distribution obtained in Fig.l0 fullydemonstrates and explains the location of severe deformation and

0.2-,--------- ----------,

0.1I::::.""C1:l~...,

if!. O.[fl[flQ)

I::::,..::.:

-0.1(j

.""..c:E-o

-0.2

-0.3 I I I I I Io 100 200 300 400 500 600 700

Distance from the Point Cl (mm)

Fig.l0 Thickness strain distributions for a front fender along theline CI-C2.

the tendency of overall deformation with an almost similarresult to that of incremental finite element analysis. Noting thatthe present algorithm provides information on an initial blankshape and strain distribution at the initial stage of part design,the present algorithm can be considered indispensable for partdesign and process design.

4. Conclusion

An algorithm has been developed for prediction of initialblank shapes and strain distributions from desired final shapes insheet metal forming process. In this approach, the material isdescribed by the Hencky's deformation theory and the Hill'sanisotropic yield criterion. A finite element formulation isderived to minimize an approximated plastic potential energywhich considers various process variables such as the frictionforce, the blank holding force, and the draw-bead force.

The numerical and experimental results of the square cupdrawing demonstrate that the present algorithm calculates initialblank shapes and strain distributions within small range of error.It is shown that the present algorithm can analyze complicatedproblems such as oil pan drawing and front fender stamping byproviding reasonable blank shapes and thickness straindistributions. Although the present approach does not describedeformation process precisely following its path, it providesuseful information on process variables at the initial design stagewith rapid calculation.

References

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(1989) 345.[4] J.H. Vogel. and D. Lee, Int. J. Mech. Sci. 32 (1990) 891.[5] X. Chen and R. Sowerby, Int. J. Mech. Sci. 34 (1992) 159.[6] R. Sowerby, J.L. Duncan and E. Chu, Int. J. Mech. Sci. 28

(1986) 415.[7] G.N. Blount and P.R. Stevens, J. Mater. Process. Techno!. 24

(1990) 65.[8] Y.Q. Guo, J.L. Batoz, J.M. Detraux and P. Duroux, Int. J.

Numer. Methods Eng. 30 (1990) 1385.[9] Y.Q. Guo, J.L. Batoz, M.EI. Mouatassim and J.M. Detraux,

NUMIFORM (1992) 473.[10] K. Chung and O. Richmond, Int. J. Mech. Sci. 34 (1992)

575.[11] K. Chung and O. Richmond, J. of Applied Mechanics. 61

(1994) 176.[12] S.Y. Chung and H.W. Swift, J. Iron Steel Inst. 170 (1952)

29.[13] S.D. Liu and M.Karima, NUMIFORM (1992) 497.(14) A. Makinouchi, E. Nakamachi E. Onate and R.H. Wagoner

(eds), NUMISHEET, Isehara, Japan, 1993.