The “internal-variables” approach in friction modeling of computer simulation in sheet-metal forming processes

Journal of Materials Processing Technology, 37 (1993) 9~114

95

Elsevier

The "internal-variables" approach in friction modeling of computer simulation in sheet-metal forming processes Tze-Chi H s u

Department of Mechanical Engineering, Yuan-Ze Institute of Technology, Neih-Li, Taoyuan, Taiwan, ROC

Industrial S u m m a r y Despite the complexity and importance of friction, most current simulations of metalforming processes use relatively simple friction models such as the Amontons-Coulomb constant friction coefficient. In the present work, a more realistic friction model that can take account of the fundamental processes parameters involved in friction and lubrication at the tooling/workpiece interface has been developed. "Internal variables" such as the lubricant film thickness, the tooling roughness and the workpiece properties are required for a realistic friction model. The calculation of these variables involves the use of traditional "external variables", such as the pressure and the material sliding velocity, from finiteelement deformation analysis. The active lubrication regime and a suitable friction model can then be determined from these local variables.

1. I n t r o d u c t i o n The p u r p o s e of l u b r i c a t i o n is to s e p a r a t e two s u r f a c e s s l i d i n g p a s t e a c h o t h e r w i t h a film of some m a t e r i a l w h i c h c a n be s h e a r e d w i t h o u t c a u s i n g a n y d a m a g e to t h e s u r f a c e s . P r o v i d i n g a sufficient t h i c k n e s s of l u b r i c a n t film h y d r o d y n a m i c a l l y is t h e k e y e l e m e n t in e l i m i n a t i n g w e a r or s u r f a c e d a m a g e . In m e t a l f o r m i n g p r o c e s s e s a n y of s e v e r a l d i f f e r e n t r e g i m e s of l u b r i c a t i o n c a n o c c u r a t t h e t o o l i n g / w o r k p i e c e i n t e r f a c e . T h e s e r e g i m e s c a n be c h a r a c t e r i z e d by t h e t h i c k n e s s of l u b r i c a n t film r e l a t i v e to t h e s u r f a c e r o u g h n e s s a n d by t h e f r a c t i o n of i n t e r f a c e l o a d t h a t is c a r r i e d by t h e c o n t a c t of r o u g h n e s s p e a k s or a s p e r i t i e s . W i l s o n [1] h a s d e s c r i b e d f o u r m a i n r e g i m e s t h a t o c c u r in t h e m e t a l - f o r m i n g p r o c e s s : t h i c k film, t h i n film, m i x e d film a n d b o u n d a r y l u b r i c a t i o n , as illust r a t e d in Fig. 1.

Correspondence to: T.-C, Hsu, Associate Professor, Department of Mechanical Engineering, Yuan-Ze Institute of Technology, Neih-Li, Taoyuan, Taiwan, ROC. 0924-0136/93/$06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.

96

T.-C. Hs u / Internal-variab les approach

Tooling

Workpiece

(a)

Workpiece (b)

Tooling

Boundary Film

Workpiece

(c) Tooling

Workpiece

(d) Fig. 1. Lubrication regimes: (a) thick film; (b) thin film; (c) mixed regime; and (d) boundary regime. Friction is most commonly characterized by using the constant coefficient of friction model attributed to Amontons [2]. In the lightly loaded condition, friction is generated by the adhesion at the contacting peaks of two generally elastic surfaces [3]. Provided the adhesion stress is not too large, the pressure

T.-C. Hsu/Internal-variables approach

97

at the contacting peaks is equal to the hardness of the softer material in the contact. Since the hardness is about three times the yield strength of the workpiece, the real area of contact would be expected to be about one third of the apparent area and hence the friction force is proportional to the normal load in accordance with the constant coefficient of friction characterization. Thus, the Amontons-Coulomb model of friction is applicable to the boundary lubrication of lightly loaded elastic contacts. However, if the pressure is sufficient to cause bulk deformation, the hardness of the material will no longer be constant and the contact area between two asperities will be changed according to the changing hardness. Wilson [1,4,5] has pointed out that such simple models cannot reflect the influence of process variables on friction. This limitation of the simple friction model greatly reduced the usefulness of computer simulation as a design tool. A more realistic friction model which can take account of the fundamental processes parameters involved in friction and lubrication at the tooling/workpiece interface has been developed by Wilson, Hsu and Huang [6]. The same strategy is adopted in the present paper by focusing on the load-sharing process in the mixed-lubrication regime, starting with the formulation of the lubrication problem in the sheet-metal forming process and then disussing the calculation of internal variables for various lubrication regimes. The friction stress occurring at the tooling/workpiece interface is then modified, depending on the conditions of asperity contact. Finally, several numerical examples are compared with the measured strain distribution in axisymmetric stretch forming. In general the results of the simulation agree well with the experimental measurements.

2. F o r m u l a t i o n of the lubrication problem Hydrodynamic lubrication in sheet-metal forming may be characterized by the Reynolds equation in both the inlet zone and the work zones [7]. The inlet zone is the wedge section immediately outside the region of apparent contact: it travels with the contact edge as deformation proceeds. When the sheet rapidly wraps around the punch, the squeeze action and the relative sliding velocity of the surfaces into the wedge at the edge of contact will entrain the lubricant into the contact region. It may be assumed that the pressure in the inlet region is determined largely by the hydrodynamics and does not affect the shape of the surfaces. On the other hand, in the contact region or work zone the pressure is determined largely by plasticity and does not influence the lubricant flow. The division of the film into the inlet and work zones greatly simplifies the process of solution.

2.1. Thick-film lubrication The first mathematical analysis of hydrodynamic lubrication, which is the foundation of subsequent lubrication theory, was given by Reynolds [8], who showed that a converging, wedge-shaped film was necesssary to generate

T.-C. Hsu/ Internal-variables approach

98

pressure in the oil film. If the process is operating under conditions of plane strain in the thick-film regime the appropriate form of the Reynolds equation is:

5-~ \12it 5x ] = ~x ( O h ) + 5--t

(1)

where h is the local film thickness, x is the distance along the film, t is the time, tt is the lubricant viscosity and O is the average surface speed. Physically this describes the variation in the pressure p with position and time in the thin film of viscous liquid lubricant between two moving surfaces. The left-hand side of eqn. (1) is the gradient in the pressure flow, which should balance the flow gradients caused by the movement of the two surfaces along the film (shear flow) and normal to the film (squeeze action). The generalized inlet zone system consisting of lubricant, rough tooling and workpiece is shown in Fig. 2 (a). R1 and Rz are the local average radii of the tooling and the workpiece, respectively, U1 and U2 are the local tooling and workpiece velocities, and h the local average film thickness. The coordinate system is fixed with respect to the boundary between the inlet- and work-zones and the system geometry is equivalent to that shown in Fig. 2 (b) if R1 R2 R=- R1 + R2

(2)

Using the traditional parabolic approximation, with the geometry shown in Fig. 2 (a), the local current film thickness in the inlet zone is given by x2

h(x) = hl (t) + - -

2R(t)

(3)

where h~ is the local film thickness. The boundary conditions for the lubricant film are x=0

P =Po

(4)

p=0

(5)

and x= ~

After substituting eqn. (3) into the governing equation (1),which assumes that the roughness effect is insignificant in the thick-film regime and applying the boundary conditions (4), (5), the lubricant film thickness in the inlet zone can be calculated as the foil bearing derived by Bolk and Van Rossum [9]. 2.2. Thin-film lubrication theory

One of the basic assumptions of the Reynolds equation is that the characteristic length in the main direction o f flow is much greater than the film thickness. Considering the actual surface roughness dimensions generated in the manufacturing process, as well as the typical film existing hydro-

T.-C. Hsu/ Internal-variables approach

~

99

h

al

R

U1

U2

b) Fig. 2. (a) Generalized inlet-zone; (b) equivalent inlet-zone geometry. dynamically in forming operations, the application of Reynolds equation would violate that assumption and make the analysis questionable. Thus, if the roughness of the surfaces is significant compared with the mean lubricant film thickness, then an average Reynolds equation which allows for the influence of roughness on lubricant flow should be used. Elord [10] has provided a useful review of the problem. Patir and Cheng [11] derived an average Reynolds equation in terms of pressure and shear-flow

T..C. Hsu/ Internal-variables approach

100

factors, which are functions of local surface-roughness and lubricant film thickness. These empirical flow factors can be obtained from the mean flow quantities by using a "patch" of surface to simulate the lubricant flow numerically with known statistical properties or measured surface roughness. Their average flow model can be written as

/

h 3 ~t9)

-~h

U1-U2

~dps . ~h

(6)

a is the composite surface roughness defined by a = ~

(7)

~bx and ~b~ are the pressure and shear-flow factors which compensate for the effect of roughness and may be expressed [12] as functions of h and a as:

¢x=1

2\h]

(8)

and 3 a o'~ + a~ q~s=2 h a2

(9)

Physically, q~x characterizes the blocking of flow by surface ridges normal to the pressure gradient (¢x < 1) or the admission of flow by roughness valleys parallel to the pressure gradient (~b~> 1). ~bs captures the tendency of a rough surface to drag lubricant in the direction of surface motion [13]. After using the same approach as outlined in previous section, the film thickness in the thinfilm regime can be calculated also.

2.3. Mixed-film lubrication In the mixed-lubrication regime shown in Fig. 1 (c) the mean lubricant film thickness is less than about three times the RMS composite roughness of the surfaces. The interface loading is shared between the pressurized lubricant film in roughness valleys and "contact" at asperity peaks. In principle, the average flow model described above may be used to calculate the average pressure p in the thin-film regime as well as the average lubricant pressure Pb of the "valley" in the mixed regime. The asperity contact pressure Pa can be calculated by a semi-empirical equation which relates the effective hardness of a plastically deforming substrate to the contact area of two asperities. The semi-empirical expression developed by Wilson and Sheu [14] may be written as H=

2

f, (A)E+f2(A)

(10)

T.-C. Hsu/ Internal-variables approach

101

where H=(pa-Pb)/k

(11)

E=

(12)

(Ul-U2)Ot

fl (A) = - 0.86A 2 + 0.345A + 0.515

(13)

and fz(n) =

1 2.571 - A - A In (1 - A)

(14)

A is the contact fractional area, U1 and U2 are the workpiece and tooling velocities respectively, 0t is the tooling asperity slope, k is the workpiece shear strength, k is its strain rate and l is the asperity half-pitch. Although most derivations of the flow factors Cx and Cs assume relatively large surface separations compared to the surface roughness and the surface roughnesses are uncorrelated, it is useful to model the mixed regime by coupling the average flow model with the asperity deformation models outlined in eqns. (10)-(14). The basic models of asperity deformation are still applicable in the mixed regime. However, the real area of contact is reduced by the presence of pressurized lubricant in the roughness valleys. A simple inlet-zone approach in sheet-metal forming processes will be used as an example of the analytical approach to modeling film formation in the mixed regime. The load is shared between the asperity contact and lubricant film. The local average pressure p in the mixed regime contains both thin-film and boundary components p =paA +Pb ( l - A ) = (Pa--pb)A +Pb

(15)

wherepa is the average asperity pressure, Pb is the average valley pressure. The quantity (Pa--Pb) is calculated by using the effective hardness theory, Pb is calculated using thin-film lubrication theory and A is calculated from the assumed roughness height distributions. The load-sharing equation (15) can be analyzed by starting to calculate the lubricant pressure Pb- The boundary conditions at time t for the thin-film Reynolds' equation are ~P=o ~x

x=0,

h=hx

P=Pl

x = oo

h = Go

P-----Pa----Pb-~0

(16) (17)

where the pressurepx at the boundary between the inlet and work zones is fixed by plasticity and equilibrium conditions. After using the parabolic approximation for the film shape, eqn. (6) can be integrated numerically to give the valley pressure Pb~ at the contact edge subjected to the boundary condition (16) and

T.-C. Hsu/ Internal-variables approach

102

(17). The resulting lubricant pressure Pbl can be represented in non-dimensional form as

fibl =tip +fis

(18)

where

f

lop=A

/is = A

x

¢-~H~f3dx+/~ ¢ ~ f 3 d x + C

o(f0

~o~

f0dx)

dx - Hll ~ ~bxH 3

fox3

~ ¢ ~ f 3 dx

(19)

(20)

and

Phi =Phi/K

(21)

The non-dimensional pressure i0p is the pressure generated in the wedge action and los is the pressure generated in the shear-flow effect caused by the anisotropic roughness. These non-dimensional parameters are expressed as

A=6v/-Ra ttV/a2 g

(22)

B = 1 2 h l # R /a2 K

(23)

C= - 2 R # / a K

(24)

/ ) = ( a 2 -G2)I a2

(25)

Hf=h/c:

(26)

H1 = h i / a

(27)

and q~x= 1 --~ H 2

(28)

The fractional contact area A1 at the boundary between the inlet- and workzones is calculated using the method given by Christensen [15] as

Al=~(~-

z + z 3 - ~ z s + ~ z 7)

(29)

where z = hi/3a

(30)

and a is the composite surface roughness. The effective hardness equation (10) is modified in Wilson's analysis [16] to include the workpiece conformation under various considerations. If quasisteady conditions with incomplete contact are assumed at the boundary between the inlet- and work-zones, the sliding model can be simplified to H=

2A1S ( l - A 1 ) fl (A1) + S(f2(A1))

(31)

103

T.-C. Hsu/ Internal-variables approach

where the non-dimensional sliding speed S is defined as S = I Ua - U210t/~l

(32)

and the functions fx and/'2 are defined by eqns. (13) and (14), respectively. The complete problem is thus solved by a " s h o o t i n g " method. F o r a fixed p~, a value of hi is assumed and t h e n the thin-film Reynolds' e q u a t i o n (19) is used to find the pressure in the valleypba and eqns. (29) and (31) are used to c a l c u l a t e A~ and H. These values are t h e n combined to c a l c u l a t e the a v e r a g e pressure Pt fit = H A 1 +/561

(33)

where the non-dimensional pressure fit is defined as (34)

fit=Pt/K

If the c a l c u l a t e d value ofpt does not m a t c h the imposed v a l u e P l , the assumed v a l u e of hi is adjusted until it does. Since the pressure carried by the l u b r i c a n t drops to zero as the film thickness decreases from the mixed regime to the b o u n d a r y lubrication, the c o n t a c t area A1 at the t r a n s i t i o n zone can be c a l c u l a t e d from eqn. (31). The c o r r e s p o n d i n g l u b r i c a n t film thickness can t h e n be solved from eqn. (29): the results are shown in Figs. 3 and 4. F i g u r e 3 is the c o n t a c t - a r e a ratio u n d e r different c o n t a c t

0.5 ~_ o ",tzt

+

~,

.t

r

; p=2.0

0.4

:::16 0.3

0

.-I

~

~

¢

;

:,

v

~

~,-'-

..-1

r'z.

.~.

r,.)

1.2

0.2 ~.~_

0.8

_

0.1 ~,-."

0.0

_"I

r_

I

I

I

I

I

10

20

30

40

50

Fig. 3. Contact-area ratio versus S.

60

104

T.-C. Hsu / Internal-variab les approach 2 --

.~

~

--I-

-~

m p=0.2

_-

~=

T=

--

i

_--

--

0.6

_

--

-

-

--

--

-

1.0

_

--

-~

-"

-"

"-

O

E

O

1.4

Z

~I

0

I-

I-

I

I-

~

1.8

~..

I

I

I

I

I

10

20

30

40

50

60

S Fig. 4. Non-dimensional film thickness versus S.

p r e s s u r e w i t h r e s p e c t to the n o n - d i m e n s i o n a l sliding v e l o c i t y S in the t r a n s i t i o n zone a n d Fig. 4 is t h e c o r r e s p o n d i n g n o n - d i m e n s i o n a l film t h i c k n e s s . T h e c o n t a c t - a r e a r a t i o is p r o p o r t i o n a l to t h e c o n t a c t p r e s s u r e a n d i n v e r s e l y p r o p o r t i o n a l to S. H o w e v e r , f r o m Fig. 3 it c a n be seen t h a t as the sliding i n c r e a s e s , the a s p e r i t y c o n t a c t will h a v e a c o n s t a n t v a l u e t h a t is i n s e n s i t i v e to c h a n g i n g S and is a f u n c t i o n of the c o n t a c t p r e s s u r e only. T h e r e l a t i o n s h i p s h o w n in Fig. 4 for t h e c o r r e s p o n d i n g n o n - d i m e n s i o n a l film t h i c k n e s s p r e d i c t s t h a t the i n c r e a s i n g c o n t a c t p r e s s u r e will h a v e a l o w e r limit for the s t a r t i n g of t h e b o u n d a r y regime. B o t h the c o n t a c t - a r e a r a t i o a n d the n o n - d i m e n s i o n a l film t h i c k n e s s s h o w n in Figs. 3 a n d 4 c a n be f o r m u l a t e d as e m p i r i c a l e q u a t i o n s in t e r m s of nond i m e n s i o n a l sliding speed S a n d n o n - d i m e n s i o n a l p r e s s u r e t5 by the m e t h o d of c u r v e fitting. T h e e m p i r i c a l e q u a t i o n for n o n - d i m e n s i o n a l c o n t a c t - a r e a r a t i o is e x p r e s s e d as

A = F1 (S) F : ( P ) where F1 ( S ) = 0 . 4 1 6 7 - 0 . 1 2 9 S + 2.927 × 1 0 - 2 S 2 - 2 . 9 7 9 x 10-3S3 + 1.109 x 1 0 - 4 S 4

(35)

T.-C. Hsu / Internal-variab les approach

105

Table 1 Constant coefficients in empirical equations (40) and (41)

D, D2 D3 D4 D5 D6

S<5

5
10
20
50
0.352 0.505 2.83 2.9324 1.3812 0.2998

0.592 0.131 2.83 2.9324 1.3812 0.2998

0.694 0.062 2.3843 2.0483 0.8212 0.1677

0.766 0.029 2.3843 2.0483 0.8212 0.1677

0.815 0.029 2.3843 2.0483 0.8212 0.1677

and

F z ( P ) = l . 2 0 8 2 x 10-2+1.2233/5

for

S<10

(36)

where F1 (S) = 0 . 2 6 8 6 - 6 . 2 9 2 × 10- 3 S + 2.843 × 1 0 - 4 S 2 - 4 . 7 4 8 x 1 0 - 6 S 3 and

F2(P)=-l.5xlO-3+O.9967P

for

10
(37)

where F1 ( S ) - 0.2172-8.8 × 1 0 - 5 S and F2(P)=-1.706×10-2+l.017P

for

20
(38)

The empirical e q u a t i o n for non-dimensional film thickness HI is w r i t t e n as H1 ---- F3 (S) F4 (P)

(39)

where

F3 (S) = D 1 S D2

(40)

and F4 (P)

= D3

-

D4 P + D5 p2

_

D6

p3

The values of coefficients D 1 - D 6

(41) are

listed in Table 1.

3. F r i c t i o n m o d e l The friction stress between the tool and the workpiece i n t e r f a c e can be divided into t h r e e parts, associated with plowing, adhesion and viscous drag. The r e l a t i v e i m p o r t a n c e of the c o n s t i t u e n t s of the friction stress is d e t e r m i n e d by a n u m b e r of factors, including the l u b r i c a t i o n regime, which can be characterized by the load-sharing proeess between the bulk l u b r i c a n t film and the

106

T.-C. Hsu / Internal-variables approach

asperity contacts. In full-film lubrication, which includes thick-film and thinfilm regimes, asperity contact is prevented by a relatively thick lubricant film which completely carries the interface pressure. In this regime, the film thickness would typically be larger than 10 times the RMS composite roughness of the tool and workpiece in the thick-film regime and between 3 and 10 times in the thin-film regime. Since no asperities contact occurs, the friction stress ~f is due to viscous resistance to lubricant shear and the friction stress can be expressed by

zf = rv = Cftt U/h

(42)

where the shear stress factor ¢f is a correction for the increase in the average viscous shear stress due to surface roughness. In the thick-film regime, the stress factor is equal to unity. As in the case of the thin-film regime, thin-film theory provides many different expressions for q~f. The expression given by Patir and Cheng [17] is , /z+l l c~f= 3~-~2z ( 1 - z 2 ) 3 m ~ - - ~ - ) + 6 0 { -55 + z[132 + z(345 + z

x {-160+ z[-40+ z(60+147z)] })] }l

h/a<3

(43)

and

cPf=3a{z[ (1-z2)31n(z+l~\~-lj +-i5z [66 + z2(30z2-80)]]

h/a>3

(44)

where

z = h/3a

(45)

and fl = 1/300

(46)

h is the local mean film thickness and a is the composite surface roughness. In most metal-forming processes with liquid lubrication, the tribological conditions are unsuitable for complete separation of the tool and the workpiece. If the film thickness is less than 3 times the RMS composite roughness of the tool and the workpiece then the interface is operating in either the mixedor the boundary-regime. In either case modeling should proceed by estimating what part of the interface p is borne by asperity contact: The results presented in Section 2.3 can be used to determine the transition process from the mixedlubrication to the boundary-lubrication regimes. In the mixed-lubrication regime, where the average film thickness is smaller, some asperity contacts are established and the load is shared partly by the asperities and partly by the pressurized film entrapped in valleys of the rough workpiece. Therefore the friction stress consists of adhesion and plowing

T.-C. Hsu / Internal-variab les approach

107

components due to asperity contacts and also the viscous resistance from hydrodynamic lubrication. The frictional stress Tf can then be written as zf = zaA + z p A + Zv(1- A)

(47)

where A is the fractional contact area determined by eqn. (29). Za, Zp and zv are the adhesion, plowing and viscous friction terms, respectively. The adhesion and plowing components are given by Za= Ck

(48)

and Tp : 0 t k H

(49)

respectively, where c is the adhesion coefficient. In the case of the boundary-lubrication regime, the load is supported completely by asperity contacts and the pressure of the entrapped film in the valley of a rough workpiece drops to zero. The viscous resistance of the lubricant flow is usually neglected and the friction stress is written as zf = zaA + zpA

(50)

Once the non-dimensional hardness H and the non-dimensional strain rate E are known, the semi-empirical equations (10)-(14) can be used to calculate the fractional contact area A in the boundary regime. Thus, if the various material properties, the average film thickness and the surface roughness are known, all the friction components can be evaluated and the friction estimated.

4. Experimental work and comparison with theory The implementation of the realistic friction model has been discussed by Wilson, Hsu and Huang [6]. At least three internal interface variables corresponding to mean lubricant film thickness, tooling roughness and workpiece roughness are required for a realistic friction model. The active lubrication regime and appropriate friction model can be determined from the current local values of these internal variables. Friction can then be calculated from the internal variables combined with the traditional external variables such as pressure and sliding speed. The present computer simulation combines the friction model and lubrication analysis described in the previous section with the finite-element code outlined by Hsu [18] to analyze the axisymmetric stretch-forming operation. Axisymmetric punch-stretching experiments were conducted by using a computer-controlled hydraulic press. Experimental variables, which include the mechanical properties and the surface topography, were measured before deformation. Several different lubricants, ranging from high viscosity to low viscosity, were applied at the tool/workpiece interface. The strain

108

T.-C. Hsu/ Internal-variables approach

distributions were measured after deformation and the experimental data were compared with the computer-simulation results. A set of die plates was added to accommodate the axisymmetric stretchforming test, consisting of a 44.65 mm diameter polished steel punch and die plates with a 50.78 mm opening. A U-shaped groove and a V-shaped draw-bead were machined into the lower- and upper-die plates, respectively. The diameter of the clamped perimeter was 62.84 mm. The workpiece samples were blanked into circular disks with a diameter of 78.30 mm. After the blanking process each workpiece sample was gridded with 2.54 mm diameter circles. The diameter of the circles in the radial direction along two sides was measured using a toolmaker's microscope to an accuracy of 0.005 mm. The measured radial strain distributions for 16 mm dome height and punch speeds of 7.62 and 2.54 mm/s are shown in Fig. 5. The workpiece surface roughness is the original sheet roughness in this case. The strain distribution at the higher punch speed is more uniform than the strain distribution at the

0.3

v

0.2'

0.1

ool 0.0

I

i

I

I

0.2

0.4

0.6

0.8

Undeformexi Coordinate / Punch Radius Fig. 5. Experimental measurement of strain distributions at different punch speeds,

.0

109

T.-C. Hsu/Internal-variables approach

lower punch speed, the reason for which is that the high punch speed tends to form a larger region in the thick-film regime at the interface, compared to the low punch speed. Thus, the strain distribution at high punch speed is flatter in the small region near the center of the workpiece associated with the essentially frictionless condition occurring in the well-lubricated region. A comparison between the simulated results and the experimental data is shown in Figs. 6 and 7, for the low-speed simulation and for the high speed simulation, respectively. In general, the simulation using the current model follows the trend of the experimental data quite well, although there is a tendency to over-predict strains from just outside the central area to near the strain peak. In fact, it looks as though the experimental strain distribution is displaced outwards in this area, relative to the predictions of the theory. This suggests that the well-lubricated region is somewhat larger than as predicted by the model, which may be due to the influence of surface roughening on lubricant transport, which is not modeled in the present simulation. On the other hand, the simulation using the constant-friction model shows a much larger discrepancy compared with the experiment s .

"d ~t = 0.2

/

0.3 v

r,o et0

Current Model

0.2

0.1 []

Experiment

0.0 0.0

0.5

1.0

Undeformed Coordinate / Punch Radius Fig. 6. C o m p a r i s o n b e t w e e n e x p e r i m e n t a n d s i m u l a t i o n r e s u l t s for t h e l o w - s p e e d case.

T.-C. Hsu/Internal-variables approach

110

0.4

Ix= 0.2

./

"d 0.3

Current Model

0.2

0.1 a

0.0

Experiment

J

0.0

0.5

1.0

Fig. 7. Comparison between experiment and simulation results for the high-speed case.

The effect of increasing speed is to increase the region of hydrodynamic lubrication. Another technique to achieve a hydrodynamic lubrication condition at the interface is to reduce the workpiece surface roughness. The tests were run with the tooling and the workpiece polished to a roughness of about 0.2 ~m Ra. The resulting different strain distribution with a punch speed of 2.54 mm/s is compared with the previous data for surfaces with a roughness of about 1 lam Ra in Fig. 8. As expected the smoother surfaces result in a more uniform strain distribution. The results obtained from the simulation for the smooth surface are shown in Fig. 9. Again the simulation predicts strains quite well, but with an apparent outward displacement of the experimental data relative to the prediction. Figure 10 shows the distribution of effective friction coefficient for all three cases, its value increasing from 0.002 in the hydrodynamic-lubrication region to 0.14 in the boundary-lubrication regime.

T.-C. Hsu / Internal-variab les approach

111

0.3

=0.2 ~tm

0.2

~

¢x0

=1.0 ~tm

e~0

m

0.1

0.0 I 0.0

0.5

1.0

Undeformed Coordinate / Punch Radius Fig. 8. Experimental measurement of strain distributions for different surface roughnesses.

5. C o n c l u s i o n s

The internal-variables friction model has been implemented successfully in the axisymmetric sheet-metal stretch forming process. The hydrodynamiclubrication region formed in the contact zone between the tool/workpiece is predicted by the cur r ent friction model. The transition between the mixed-film regime to the boundary regime is also performed by considering the loadsharing process between the asperity contact and the lubricant carrying capability. The semi-empirical equations of the contact-area ratio are obtained for different sliding velocity and interface pressure. The friction stress can then be

112

T.-C. Hsu/Internal-variables approach

0.4

~t = 0 . 2

/

0.3 I

=

•

. rrent Model

0.2' ~0

0.1 []

0.0 0.0

Experiment

!

!

0.5

1.0

Undeformed Coordinate / Punch Radius Fig. 9. Comparison between experiment and simulation results for smooth case.

evaluated by combining these internal variables with those traditional external variables from the finite-element method such as the sliding speed and the contact pressure at each node. The comparison of the strain distribution between the simulation results and the experiments shows the superiority of the current friction model over the constant-friction model. The computational time is increased by less than 15% in using the internal-variables friction model. This relatively small penalty for a significant improvement in predictive capability should increase the likelihood t h a t the results of this research will be adopted widely.

Acknowledgements The author wishes to t h a n k the National Science Council of Taiwan for the financial support of this research by grant NSC 81-0401-E-155-01. The computing facility provided by Yuan-Ze Computing Center is also greatly appreciated.

113

T.-C. Hsu/ Internal-variables approach

0.2 I

Rough, Low Speed

f o ..el 0.1

~

~9

Smooth, Low Speed

O

L~

0.0 ~ 0.0

0.1

~ 0.2

0.3

0.4

0.5

0.6

Undeformed Coordinate / Punch Radius Fig. 10. Effective friction coefficient distribution.

References [1] W.R.D. Wilson, Friction and lubrication in sheet metal forming, in: D.P. Koistinen and N.M. Wang (Eds.), Mechanics of Sheet Metal Forming, Plenum Press, New York 1978, pp. 157-177. [2] D. Dawson, The History of Tribology, Longman Group Ltd., London, 1979, pp. 99-100; 154-156 and 21:~222. [3] F.P. Bowden and D. Tabor, The Friction and Lubrication of Solids, Oxford Clarendon Press, Oxford, Vol. 1, 1953, pp. 98-100. [4] W.R.D. Wilson, Friction and lubrication in bulk metal forming processes, J. Appl. Metalwork., 1 (1979) 1 19. [5] W.R.D. Wilson, Friction modeling in forging, in: R. Arseuault (Ed.), Computer Simulation in Material Science, ASM, New York, 1987, pp. 237-267. [6] W.R.D. Wilson, T.C. Hsu and X.B. Huang, A realistic friction model for computer simulation of sheet metal forming process, ASME, Winter Annual Meeting, Atlanta, 1991. [7] W.R.D. Wilson and J.J. Wang, Hydrodynamic lubrication in simple stretch forming processes, ASME J. Tribol., 106 (1984) 70-77.

114

T.-C. Hsu / Internal-variab les approach

[8] O. Reynolds, On the theory of lubrication and its applications to Mr. Beauchamp Tower's experiments, Philos. Trans. R. Soc., 177 (1886) 157-234. [9] H. Blok and J.J. Van Rossum, The foil bearing - A new departure in hydrodynamic lubrication, Lubr. Eng., (1953) 316. [10] H.G. Elord, A review of theories for the fluid dynamic effects of roughness on laminar lubricating films, in: D. Dawson et al. (Eds.), Proc 5th Leeds-Lyon Symposium on Lubrication, 1978, pp. 11 26. [11] N. Patir and H.S. Cheng, An average flow model for determining effects of threedimensional roughness on partial hydrodynamic lubrication, J. Lubr. Technol., 100(1) (1978) 12-17. [12] J.H. Tripp, Surface roughness effects in hydrodynamic lubrication: the flow factor method, J. Lubr. Technol., 105(3) (1983) 458-465. [13] W.R.D. Wilson, Mixed lubrication in metalforming processes, Advanced Technology of Plasticity, 1990, pp. 1667-1677. [14] W.R.D. Wilson and S. Sheu, Real area of contact and boundary friction in metal forming, Int. J. Mech. Sci., 30(7) (1988) 475-489. [15] H. Christensen, Stochastic models for hydrodynamic lubrication of rough surfaces, Int. J. Mech. Eng., 104 (1970) 1022-1033. [16] W.R.D. Wilson, Friction models for metal forming in the boundary lubrication regime, in: I, Haque et al. (Eds.), Friction and Material Characterization, ASME, New York, 1988, pp. 13-23. [17] N. Patir and H.S. Cheng, An application of average flow model to lubrication between rough sliding surfaces, J. Lubr. Technol., 101(2) (1979) 22(~230. [18] T.C. Hsu, The friction modeling of computer simulation for sheet metal forming process, Ph.D. Dissertation, Northwestern University, Evanston, IL, June, 1991.