Int..I. Mech. Sci. Vol. 24, No. 8. pp. 479-493, 1982 0020-7403/82/080479-15503.00/0 Printed in Great Britain. Pergamon Press Ltd.

FINITE E L E M E N T ANALYSIS OF METAL FORMING PROCESSES WITH ARBITRARILY SHAPED DIES

S. I. OH Battelle Columbus Laboratories, 505 King Avenue, Columbus, OH 43201, U.S.A.

(Received 23 May 1981; in revised form 21 September 1981)

Summary--A method of discretizing the die boundary conditions is considered for the analysis of metal forming processes by the rigid viscoplastic finite element method. The method is natural to the finite element method and general enough to treat the boundary conditions of arbitrarily shaped dies in a unified way. Solutions of the spike forging process are obtained by using the method. The deformation mechanics of the spike forging process are discussed and the solutions are compared with experiments.

N O T A T I O N

v~ velocity component iij strain rate tensor component

effective strain rate effective strain

cr~j stress tensor component ~0' deviatoric stress component

effective stress 1¢ prescribed boundary traction U prescribed boundary velocity f frictional tration

q~ variational functional E work function

Hi distribution function for node i ui node velocity

vo die velocity n die surface unit normal s die surface unit tangent

K penalty constant /z viscosity x material coordinates

m friction factor k local shear yield

Ro initial workpiece radius H0 initial workpiece height VD die velocity (IVo[)

I N T R O D U C T I O N

A major objective of mathematical modeling of metal forming processes is to provide necessary information for proper design and control of these processes. Therefore, the method of analysis must be capable of determining the effects of various parameters on metal flow characteristics. Furthermore, the computation efficiency, as well as solution accuracy, is an important consideration for the method to be useful in analyzing metalworking problems. With this viewpoint, the rigid plastic (and rigid viscoplastic) finite element method has been most successful in analyzing a wide class of metal forming problems.

The rigid plastic finite element method (FEM), which was developed by Lee and Kobayashi [ 1], has been applied to the analysis of various problem s such as solid cylinder upsetting, ring compression, extrusion, and sheet bending [1-7]. The method has been also successfully applied to the prediction of defect formation in upsetting[8] and extrusion [9], and the measurement of die-workpiece friction [ 10]. It has been extended to rigid viscoplastic material by Oh et al. [11] and applied to upsetting and ring compression in the hot working range.

The major advantage of the finite element method is its ability to generalize. That

479

480 S.I. OH

is, the method can be applied to a wide class of boundary value problems without restrictions of workpiece geometry. This is achieved by the proper discretization procedure used in the finite element method. Metal forming processes, in general, are characterized by loading with dies of various shapes. It seems that a method of discretizing the die boundary conditions, which is comparable to that used in the finite element method, is necessary in order to fully utilize the advantage of the finite element method in metal forming analysis.

In the present paper a method is proposed to establish the theoretical foundations of the rigid viscoplastic finite element analysis of metal-forming precesses with arbitrarily shaped dies. Then the solutions of the spike forging process are obtained by u s i n g t h e p r o p o s e d m e t h o d .

R I G I D V I S C O P L A S T I C F I N I T E E L E M E N T F O R M U L A T I O N The rigid viscoplastic material is an idealization of an actual one, by neglecting the eleastic response.

The material shows the dependence of flow stress on strain rate in addition to the total strain and temperature. Also it can sustain a finite load without deformation. The rigid viscoplastic material which was introduced for the analytical convenience simplifies the solution process with less demanding computational procedure. Moreover, it seems that the idelaization offers excellent solution accuracies due to the negligible effects of elastic response at large strain in the actual material. Because of this reason, the rigid viscoplastic material is especially suitable for metal forming analysis at elevated temperature. In this section a finite element formulation of rigid viscoplastic material is described. The discussions are limited to the two- dimensional isothermal deformations.

Consider a body of volume V at a generic moment with the traction F prescribed on a portion of the surface, Sv, and the velocity U on Sv. Let Sc be the remainder of the surface where the frictional stress f acts. The deformation of the body V is characterized by the following field equations.

(1) Equilibrium conditions, neglecting the body force;

wq.j - 0 (I)

where ~r~j is a stress component and "," denotes the differentiation. (2) Strain rate-velocity relation;

• 1 e~j = ~ (v~j + vj,~) (2)

where ~q and v~ are a strain rate component and a velocity component, respectively. (3) Constitutive relation;

2 i T . o,~j = 7_. e~J (3)

3e

where tTi~' is the deviatoric stress component, and # and ~ are defined by X/(3/2~rU%r0') and X/(2/3~i:~0) respectively. The flow stress #, in general, is a function of total strain, strain rate and temperature.

(4) Boundary condition;

a,jni = Fj on SF

vi = Ui on Sv (4)

JLJ = given, sign(/,) = - sign(Av,) on Sc

where n~ is a component of unit normal to the surface, and Av, is slipping velocity. The field equations (I)-(4) can be put into variational principle as

I "* 2 v~ 8*=8[f:<'*)dV+L~K'"dV--fsc{ f Ldv.}dS-fsF~V~d$]=O (5)

where the work function E(~) can be expressed as[12]

E(~) = ~0 ~ ~ d~. (6)

Here K is a large positive constant which penalizes the dilatational strain, and "*" denotes the restriction of the velocity fields to the trial space.

Finite element analysis of metal forming processes with arbitrarily shaped dies 481

The discretization of this functional follows the standard procedure and can be found elsewhere[13]. The distribution function H, which is assigned to each node, is introduced such that

U(x) = ~. H~(x)U~ (7)

V(x) = ~,/-/~(x) Vi

where U(x) and V(x) are the velocity components in x~ and x2 direction, respectively, U~ and V~ are those at nodal points, and the summations are done over nodal points. In the present investigation 9-node isoparametric elements are used.

Substituting equations (6) and (7) into equation (5), the variational functional ~ becomes nonlinear algebraic equations with unknown u,'s which are velocity components of the nodal points. The stationary point of the functional can be obtained by solving the simultaneous equations

a~P (ul . . . . . U2M) = 0, i = 1 . . . . . 2M (8) Oui

where u = ( u l , u2 . . . . . u2M-, ,u2u)=(Ul , V1 . . . . . UM, VM), and M is the number of nodal points. The solution of the highly nonlinear algebraic simultaneous equations are obtained iteratively by the Newton- Raphson method. The linearized form of equation (8) near the assumed velocity field u0 becomes

00] + r 02~ ] • Auj = O. a-~J.=,~ Laujau~ Ju=mo (9)

Solving equation (9) with respect to Auj, the assumed velocity field is updated by ui + aAu~, where a ~< 1. The iterative process continues until the fraction of the correction term, I[Aul[/llul[ and the calculated nodal point force error norm reach a preassigned value.

In certain cases of rigid viscoplastic deformation, a portion of the workpiece remains rigid. It can be proved that the rigid zone is determined uniquely by the field equations if the work function E(~) is convex. The variational function tb in equation (5) becomes non-analytic at the solution when the rigid zones are involved and the Newton-Raphson method cannot be used for the solution procedure. The difficulty has been overcome by assuming the material to be linear if ~ becomes smaller than a small positive number ~0 [11]. The rigid viscoplastic material in this approach is considered as a limiting case when Co becomes infinitely small.

In the finite element method, it is well known that the velocity field given by equation (7) is not flexible enough to provide adequate solutions for incompressible or nearly incompressible deformation. Various suggestions have been made in order to overcome this difficulty in analyzing incompressible linear elastic material[14, 15] and the elastic-plastic material in fully plastic range[16]. Numerical tests show that the reduced integration for volume strain component gives best results for the present investigation. That is, a 3 x 3 integration formula is used for distortional energy rate, while a 2 x 2 integration is used for volumetric strain rate with quadratic elements. In relation to the incompressibility, the strain rates and stresses were evaluated at the reduced integration points for best results.

B O U N D A R Y C O N D I T I O N S AND A R B I T R A R I L Y S H A P E D DIES The implementation of a regular boundary condition on SF or Su is straightforward and can be found

elsewbere[13]. In the practical analysis of metal forming processes by the finite element method, a particular attention must be paid to die boundary conditions. The frictional stress, in general, changes its direction at the "neutral point", but the location of this point is not known a priori. The "neutral point" problem has been considered by various investigators[3,4, 17] in their analyses of the ring compression test.

The shapes of dies used in metal forming processes change considerably from process to process. In the finite element analysis of metal forming processes the individual implementation of the die boundary conditions for a particular shaped die requires a substantial amount of programming effort. Therefore it is desirable to use a technique which can be applied without the restrictions of die geometries. Thus, the method becomes a practical and economical tool for the metal forming analysis. Such a technique, which is natural to FEM, is described in the present paper. This technique allows the systematic implementation of frictional boundary conditions in a unified way for arbitrarily shaped dies. It also provides the capability of predicting the neutral point location at the arbitrarily shaped die-workpiece interface. In the present investigation the constant shear friction law is employed which is a common assumption in metal forming analysis. The die boundary conditions along curved die-workpiece interfaces have been considered in the framework of FEM by several investigators [7, 18-21]. Their main emphases were on the effects of rotation on the boundary conditions in relation with the definition of the traction.

Consider, at a generic moment, a curved die surface which is in contact with workpiece as shown in Fig. l(a). The die boundary condition at the interface can be given by

V. = v D . n

Avs 2 mk tan_l ( _ ~ ) (10) f , = _ m k - ~ , ~ =

482 S.I . OH

Die

Die S,rfoce/I ~loeu~:: y "x >

i - -

Ca) (b]

FIG. 1. Schematic diagram of a curved die and a workpiece.

where n is unit normal as shown in the figure, and the subscripts n and s denote the normal and tangential direction to the interface, respectively. The expression of f, in equation (10) has been used for the smooth transition of frictional stress near the neutral point for the flat die surface [4]. Here Av, is slipping velocity, m friction factor, k local flow stress in shear, and Uo is a very small positive number compared to Av,.

The implementation of equation (10) for a curved die is not straightforward since the die surface does not coincide with the workpiece surface defined in FEM approximation. These discrepancies are shown in Fig. l(b). Because of the discrepancy, the necessary quantities in equation (10) are not well defined except at the contact node.

In order to overcome the difficulties the boundary condition normal to the surface is enforced at the contacting node and is given by

V n = v D • n~ i )

where n~i) is the unit normal to the die surface, rather than element surface, at node i. It is assumed that the slipping velocity Av, can be approximated by

Avs = ~ HiAv.i = .~, H,-(v~i-vo' s.)). ( l l )

Here, Hi is the FEM shape function on the surface, v~i the tangential velocity of the ith node, and s(0 the unit tangent to the die surface at the contact node L

By substituting equation (11) into fs expression, the integrand of the die boundary term in equation (5) can be evaluated for the two-dimensional case as

f~' fs dr, mkHj |(v,i dvo. (12) = f ' - Z - tan-' t - VD ' so,)H, ] r

q 7r /~0

The integration of the die boundary term in equation (5) is carried out by Gaussian quadrature considering the element surface as a one-dimensional isoparametric element. A higher order integration is necessary in order that the term can reflect the accurate behavior of fs near the neutral point where the integrand has large higher order contributions than that of the shape function. Numerical tests show that a five-point integration formula is sufficient for this purpose with a quadratic element.

In a case when a portion of the element boundary is not in contact with the die, the slipping velocities at the free nodes are not defined in equation (12). This difficulty can be overcome easily. The tangential die velocity vD - %) is set to zero for the free node and the small positive number ue is replaced by a very large number for the free boundary position. This procedure ensures the consistency of equation (12) on the contact portion and makes the rest traction free.

The present formulation of the die boundary conditions becomes the same as that of Chen and Kobayashi [4] in the case of the flat die surface with vD • s = 0, except that the present investigation employs quadratic elements and surface integration is done numerically. It can be easily shown that the error introduced by the present approach is of the same order of magnitude as that introduced in the FEM calculation. Therefore, the present formulation does not effect the accuracies of the FEM analysis.

I N I T I A L G U E S S The Newton-Raphson method, used in solving the nonlinear equation (8), is known to be a fast-

converging solution process. A drawback of the Newton-Raphson method is that it requires an initial guess

Finite element analysis of metal forming processes with arbitrarily shaped dies 483

for the velocity field as it can be seen in equation (9). Even though the initial guess velocity field does not affect the solution itself, it should be close enough to the solution to achieve the convergence in the iteration process. The initial guess, for the first step solution in general, can be provided by a simple upper bound solution. It seems, however, that the systematic means of providing the initial guess is desirable for the effective application of the rigid viscoplastic FEM to complicated metal forming processes. In the present investigation the initial guess was obtained by treating the workpiece as linear viscous material with the viscosity as a function of the material point.

Consider the constitutive equation of linear viscous material

~ ' = 2/.*~i~ (13)

where/., is viscosity. Comparing this with equation (3) it can be seen that the solution of rigid viscoplastic material becomes identical to that of the fictitious linear viscous material with the viscosity

_. _ I___~ (14) = ~(*) - 3~

where ~ is the flow stress of the material. The rigid viscoplastic solution can be obtained by searching for the fictitious material which satisfies the

equation (14). under an equilibrium state. This can be done iteratively by updating/~ --/~(~) using equation (14) until/~(~) does not change. This method of solution had been used by Lyness et al. [22], in their finite element analysis of non-Newtonian fluid. Comparing with the Newton-Raphson method, this method replaces the initial guess velocity field by the initial viscosity distribution. The iteration scheme with this method converges fast during the early stage of iteration and it is insensitive to the initial viscosity distribution. The convergence of this method is slower, especially near the solution, than that of the Newton-Raphson method.

In the present formulation, advantages of both methods were utilized to obtain the best solution efficiencies. That is, the Newton-Raphson method was used for the solution process, while initial guesses were generated by the method described above. Numerical tests show that a "good" initial guess can be obtained within 2--4 iterations using a uniform initial viscosity distribution throughout the workpiece. The generation of initial guess by the method is only necessary for the first step solution. For the subsequent steps, the previous step solutions are used as initial guesses.

S P I K E F O R G I N G A N A L Y S I S In order to validate the present formulation, solutions of the spike forging process were obtained. In

spike forging, a cylindrical billet is forged in an impression die containing a central cavity. The deformation characteristics of the spike forging are such that the portion of the material near the outside diameter flows radially, while the portion near the center of the top surface is extruded forming a spike. It has been reported [24, 25] that the material flow, which is characterized by spike height variation, depends on the interface friction as well as the geometries of dies and billets used.

The deformation mechanics of spike forging processes have been studied by several investigators [23- 27]. Jain et at. [24] analyzed extrusion forging using a flat top die with a sharp cornered hole by the upper bound method. Their analysis successfully explained the three distinctive modes of deformation found in experiments. Those modes of deformation consist of the stages where the spike height is reduced with increasing deformation, and the spike height remains constant until it finally increases very rapidly. Newnham and Rowe [25] also considered the flow behavior of spike forging process using a flat die with a sharp cornered hole. They analyzed the process using the slip line theory and explained the deformation mechanics. Price and Alexander[26] analyzed the spike forging process by flat platen die with a central orifice. In their analysis only the sticking die boundary condition was considered. Aitan et al. [27], analyzed spike forging process with tapered top and bottom dies by the modular upper bound method.

Computational conditions and procedure Analysis of isothermal spike forging was performed for two different frictions. The friction factors used

in the analyses are m = 0.3 and m = 0.6. The first value, m = 0.3, seems to represent an approximate value of the friction factor in actual hot forging operations under lubricated conditions [28, 29]. The second value, m = 0.6, is also a representative number in hot forging without lubrication[30]. The undeformed circular billet has the dimension of 57.2 mm (2.25 in.) in height and 50.8 mm (2.0 in.) in diameter with chopped corner to secure the stable initial position. The material used in the analysis is a +/3 microstructure Ti-6242-0.1S~ at 954 C (1750 F)[31]. The stress-strain rate relation is shown in Fig. 2. The shape and the dimension of the spike forging dies are shown in Fig. 3. The velocity of the upper die used for the analysis was 25.4 mm/sec (l.0in./sec), and the bottom die is stationary. The non-steady state deformation of spike forging was analyzed in a step-by-step manner by treating it quasilinearly during each incremental deformation. The incremental displacement of upper die for each step is chosen to be 0.02 times of the original workpiece height.

The measure of convergence represented by IIAull/llull was chosen to be 0.00001.

Results and discussions Fig. 4 shows the undeformed FEM grid used for analysis and the calculated grid distortions at die

displacements of 0.2, 0.4 and 0.58 Ho. For better comparison the grid distortions of both frictional cases are superimposed on each other. As it can be seen from the figure, the general deformation behavior of spike forging for the two different cases is similar. The solution shows that at an early stage, the deformation is concentrated near the right upper corner of the workpiece where it touches the top die, while the rest of the workpiece undergoes virtually no deformation. Because of this concentration, folding of the top surface

484 S.I . 0 8

35 0

300

25.o /

2 0 0 ~ ~ -

I0 .0 --

. 5 . 0 - -

0 . ¢ I 0.0 4.0

J

250

200

150

¢n in

I00 ~

I I I 0 8.0 12.0 16.0 20.0

Strain Rate, s e c - i

50

FIG. 2. Stress-strain rate relation of a +//microstructure Ti-6242-0.1Si at 954 C (1750 F).

Top Die

2 3. i mm

9.Smm ~1

2 5 . 4 m m i=.[ ~ o

Y/'JJJJJ///////.>" Bottom Die

FIG. 3. Schematic drawing of spike forging dies.

Finite element analysis of metal forming processes with arbitrarily shaped dies 485

"¢- u~ -c" I

.2-

o

.2 i

m u~

o.o l.o 2'.o ko :.o 2'.o RADIUS/R o RADIUS/R o

(a) (b)

0 0

~ -r" m u~

o - . : ,

d - d "

i t 0 . 0 l .O 2 .0 0 . 0 l'.O 2'.0

RADIUS/R o RADIUS/R o

(c) (d)

FIG. 4. Calculated FEM grid distortions at die displacements of (a) 0.0 Ho, (b) 0.2 Ho, (c) 0.4 Ho and (d) 0.58 Ho. Solid lines are for m = 0.6 and dotted lines for m = 0.3. Units are

multiples of the undeformed radius.

starts to take place during the early stage. As the deformation progresses, the deforming zone spreads throughout the workpiece, forming the side surface bulge. Then folding from the side surface to the top and the bottom dies takes place. Throughout the spike forging process the material near the center of the upper surface undergoes virtually no defromation as can be seen from the grid distortions. It can also be seen from the figure that the spike heights are not affected by the friction during the early stages of deformation. The figure shows, however, that higher spike formed with higher friction and that the difference in spike heights with different frictions is large at the die displacement of 0.58 Ho.

The predictions are compared with the experimentally measured spike cross sections. The experimental measurements used for comparison were obtained by Altan et at. [27], who carried out spike forging tests with lead at room temperature. These data were selected for an initial evaluation of the present FEM because they were readily available. Lead being a strain-rate dependent material, it is expected to simulate the flow of the other strain-rate dependent materials, at least approximately. This point is discussed later again in connection with Fig. 10.

The predictions with m = 0.3 are compared with lubricated case at the die stroke of 0.58 Ho in Fig. 5. In

486 S.I. OH

l-=- I L9

I - - Theory at 58 %

x~.,er ime nt oi 59%

°

0.0 1,0 2,0

RADIUSIR o

FIG. 5. Predicted cross sections compared with experiment at die displacement of 0.58 Ho. Prediction: Ti-6242-0.1Si, 954 C (1750 F), m = 0.3. Experiment: Lead, Room Temperature,

Lubricated.

Fig. 6, the predicted cross sections are superimposed with those of the dry friction experiments at die displacements of 0.3, 0.44 and 0.58 Ho. These figures show that the predictions are in excellent agreement with experiments, even though the effects of the different material behaviors of lead and titanium at their hot working range on the spike height are not known precisely. This point should be further investigated.

The details of the solutions are examined in terms of loads, velocity fields, neutral point locations, changes in spike height, and effective strain distributions. The calculated load displacement relations are shown in Fig. 7 for both frictions. The results reveal that the load increases slowly during early stages of deformation and then it increases rapidly as the deformation progresses. Similar trends were found in spike forging experiments with lead and aluminum[27]. The load values for both frictions are almost identical at the early stages and the load is higher for the higher friction.

~ / ~ ~ " Undeformed

i

(a)

- - Theory ot 50 % - - - - Experiment at 50 %

(hi

- - Theory ot 4 4 %

(C (d)

FIG. 6. Predicted cross sections compared with experiments at die displacements of (a) 0.0Ho, (b) 0.3 Ho, (c) 0.44Ho and (d) 0.58 Ho. Prediction: Ti-6242-0.]Si, 954 C (1750 F),

m = 0.6 Experiment: Lead, Room Temperature, Dry Friction.

- - Theory at 5 8 % - - ~ ~ Experiment at 59%

Finite element analysis of metal forming processes with arbitrarily shaped dies

300

0 0 0 x

2 0 0

- - ~ - - rn : 0 . 3

- - o - - r n = 0 . 6

7 x

j

J x / x /

I 0 0 x /

x ~ X / x ~ x ~ X ~

x _ x ~ x ~x

0 I I O.O 0 . 5 1.0

D i s p l o c e m e n l / R o

1 .25

I .OO

0.75

z

0.50

- 0.25

5oo

FIG. 7. Calculated load-displacement relations when m = 0.3 and m = 0.6.

487

Figs. 8 and 9 show the local velocity distributions at the die displacements of 0.02, 0.2, 0.4 and 0.58 Ho for m = 0.3 and m = 0.6, respectively. The velocities shown in the figures are relative quantities with respect to the upper die. It can be seen from the figures that the deformations at the initial stages of both frictional cases are concentrated near the right upper corner of the workpiece, while the rest of the portion is rigid. The relative velocities of spike tip are 0.188 Vv and 0.204 Vv at the die displacements of 0.2 and 0.4Ho, respectively when m =0.3, and they are 0.193 VD and 0.238 Vv when m =0.6. Here, VD is the downward velocity of the upper die. These results show that the differences in spike tip velocities with different frictions are relatively small at early stages. The figures show, however, that the flow pattern changes significantly at a die displacement of 0.58 Ho. At this stage the relative velocity of spike tip is about 1.25 Vo when m = 0.6 while it is 0.465 VD when m = 0.3.

As it can be se~n from the relative velocity patterns, a part of the workpiece flows into the central cavity while the rest of the material flows radially. Because of these two different flow directions, a neutral point forms on the upper die workpiece interface where the relative movement between die and material becomes zero. Figs. 8 and 9 show the locations of neutral points. In the case of low friction the neutral point does not form at die displacements of 0.02 and 0.2 Ho, suggesting that the material flows outward at early stages of deformation. As the deformation progresses the neutral point appears near the fillet of the upper die.

Comparing Figs 8 and 9 it can be seen that the neutral point forms earlier for higher friction and that its location is farther away from the center axis. In spite of the differences in the locations of the neutral points with different frictions, the relative movements are small along the workpiece-upper die interface for both frictions up to 0.4 Ho die displacement.

Fig. I0 shows the change of spike height as a function of die displacements. It can be seen from the figure that the top surface of the spike moves downward for both friction cases during early stages of deformation. The downward speed slows down and then the spike top surface moves up near the end of the process for both cases. The spike tip reaches its minimum earlier for the higher friction case. The figure shows that the minimum height of 34.8 mm (1.37 in.) is reached at die displacement of 31.5 mm (1.24 in.) in case of m = 0.6, while that of 31.0 mm (1.22 in.) is reached at die displacement of 35.6 mm (1.40 in.) when m = 0.3. The figure also reveals that the effect of friction on the spike height is minimal during the early stage of deformation. The variations of spike height shown in Fig. 10 are in general agreement with earlier experimental observations [24, 25].

In Fig. 10 the calculated spike height variations are compared with experimental data which were obtained from isothermally forged lead and aluminum specimens with and without lubricant. The lead was forged at room temperature and the aluminum at 31YC. It is interesting to note that "hot" aluminum and "cold" lead behave similarly during isothermal forging. Fig. 10 shows that the spike heights measured from both lead and aluminum specimens without libricant match the prediction with m = 0.3. The predictions of the friction factor, m, may require further study because of the uncertainties in spike height variations for the different materials involved. However, if it is assumed that the spike height variation is relatively insensitive to the differences in material properties involved, it can be said that the friction factor is approx. 0.6 in the case of dry friction, and it is about 0.3 in the case of lubricated forging of lead and aluminum.

MS Vol. 24, No. 8--C

488 S.I . OH

vl

ill

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RADIUS/R o (c)

o a.

'~ / t. t. z zj''t , . , - - - _ ' ~

0 . 0 1 .0 ;I.O

RADIUSIR o (d)

FIG. 8. Relative velocity distributions at die displacements of (a) 0.02 Ho, (b) 0.2 Ho, (c) 0.4 Ho and (d) 0.58 Ho when m = 0.3. A denotes the location of neutral point.

Finite element analysis of metal forming processes with arbitrarily shaped dies 489

~D

,4'

o

d "

~ 1.0

8 #

l - - -'r"

l l l / ~

I I I I I I

1.0 RADIUSIR o

(a)

0.0 l~.O

t

I l a I # • /

o I I 1 1 1 1 / ,_,

I t # / / / t

.~ l l l / l l l 11[tltl/ : t T t l l t l

o.o i'.o RADIUS/R o (b)

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l i i i

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RADIUS/R o (c)

o

t

: tt

O.O

| J • o . o "

;.o i.o RADIUS/R o (d)

FIG. 9. Relative velocity distributions at die displacements of (a) 0.02 Ho, (b) 0.2 Ho, (c) 0.4 Ho and (d) 0.58 Ho when m = 0.6. ~ denotes the location of neutral point.

490 S.I . OH

3.0

Z.¢

t.O

- - Ca lcu la ted m = 03 . . . . . Ca lcu la ted m = 0 . 6

Expe r imen t

• Lead Wi th no L u b r i c a n t o Lead Wi th F e l - P r o - C 3 0 0 • AI Wi th no L u b r i c a n t

t, AI Wi th F e I - P r o - C 3 0 0

o o i I

0.5 1.0

Displacement IRo

F1G. 10. Variation of spike heights when m = 0.3 and m = 0.6.

1.5

In the finite element method the local information such as strains, strain rates, and stresses can be easily obtained. The information can be used not only to investigate the effect of flow stress on the overall deformation but also to predict defect formation in metal forming processes. In the present study the total effective strain concentrations are shown in Fig. 11 for both friction cases at die displacements of 0.2, 0.4 and 0.58 Ha. As it can be expected from the previous discussions, the general patterns of strain distributions are similar to each other, showing almost no deformation near the spike tip and the highest strain concentration along the band joining the right upper corner of the specimen and middle of the height near the center. It is, however, interesting to note that the strain gradient is higher for the higher friction case, and that the deformation is more uniform in the case of lower friction. The difference in the strain gradient is more evident near the bottom surface of the workpiece.

C O N C L U S I O N

A method is established to discretize the die boundary conditions in the frame- work of rigid viscoplastic finite element method. The method is natural to FEM and general enough to treat the arbitrarily shaped dies in a unified manner. Improvements are made on the computational efficieneies of rigid viscoplastic FEM by introducing an automated initial guess generation scheme. It seems that the discretization of die boundary conditions, together with the initial guess generation scheme, is an essential step toward the development of a general purpose FEM code for metal forming analysis.

The present work is now being expanded to simulate metal flow in relatively complex two-dimensional metal forging applications. It is expected that this future effort will, eventually, allow the design of preform shapes, involving massive forming operations in industrial production processes.

Solutions of the spike forging process with two different frictional conditions are obtained by the present method. The deformation mechanics of the spike forging process and effects of friction on the process are discussed. The solutions are

Finite element analysis of metal forming processes with arbitrarily shaped dies 491

I i

i O"

~______ 0.?.5.~ O.ZS --'~ /

* 0

'% l I (a)

O.Z~--~"

i (d)

(b)

S "° o8

(c)

(e)

O.~

i (f) FIG. 11. Effective strain distributions at die displacements of (a) 0.2 Ho, (b) 0.4 Ho and (c)

0.58 Ho when m = 0.3 and (d) 0.2 Ho, (e) 0.4 Ho and (f) 0.58 Ho when m = 0.6.

492 S.I. OH

c o m p a r e d with expe r imen t s . The c o m p a r i s o n shows that they are in exce l len t a g r e e m e n t with each other.

Acknowledgements--This paper is based on information developed in Processing Science Program spon- sored by the Air Force Wright Aeronautical Laboratories, AFSC, under contract F33615-78-C-5025, with Dr. Harold L. Gegel as program manager. The author would like to acknowledge gratefully the helpful discussions with Professor Shiro Kobayashi of the University of California, Berkeley. The author also would like to express his thanks to Drs. T. Altan and G. D. Lahoti at Battell's Columbus Laboratories for their encouragement and suggestions during the course of this investigation.

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