An eulerian elastoplastic FEM solution to drawing process

Journal of

Materials Processing Technology ELSEVIER

J. Mater. Process. Technol. 45 (1994) 311-316

An Eulerian elastoplastic FEM solution to drawing processes Z. Malinowski

Akademia G6rniczo-Hutnicza, A1. Mickiewicza 30, 30-059 Krak6w, Poland

An elastoplastic finite element method solution to drawing processes is presented. A steady state nature of these processes allows the problem to be formulated in the spatial reference frame. The solution is divided into two variational problems. The first part involves minimization of the deformation power and gives a kinematically admissible velocity field. The second part is an integration of the stress transport equations. A numerical algorithm based on the non-linear optimization with constraints is employed to integrate the objective stress rate in the final configuration without linearization. The predictive capabilities of the method are demonstrated for drawing of a copper bar.

1. INTRODUCTION In metal forming operations such as drawing a material deformed is exposed to large elastoplastic strains. An accurate determination of the effect of process parameters on the field variables such as temperature, strain rate, strain and stress is therefore of prime interest in the process design and control. For instance, in a study of ductile damage the detailed knowledge of the strain and stress field is required. While the strain field can be successfully computed using the rigid plastic flow formulation of the finite clement method [1], the stress field predictions are not so promising. A rate elastoplastic approach is required to solve problems involving large deformations in metal forming processes. However, there are two major sources of error in the rate formulation. The first one is concerned with a proper choice of the objective tensor-rates in constitutive equations [2]. Detailed study conducted by Dubey et al. [2,3,41 has led to the formation of a family of objective stress-rates. It has been shown on a simple shear problem [4] that the use of any rate from the proposed family leads to solutions which are independent of the rate chosen. The second source of error lies in the technique used to integrate the stress-rate equations. The explicit Euler approximation violates the requirement of objectivity for rotations over a finite time step [5]

and is therefore unsatisfactory. Several methods have been proposed for time discretization in non-steady elastoplastic problems. From among them an exponential approximation of rotations [5] or a large strain velocity approach [6] seems to be adequate for metal forming processes. However, in the case of drawing processes a steady state nature of these processes can be exploited to avoid linearization or approximate solutions. The finite element schemc developed by the author and Lenard [71 for the elastoplastic analysis of the steady state flat rolling is followed here.

2. CONSTITUTIVE F O R M U L A T I O N The flow problem of an isotropic, elastoplastic material, throughout the control volume V is formulated in the spatial reference frame x~. Following the principal axes technique [2,31 the Eulerian rate of stress E

E

E

is used to construct the rate tbrm constitutive model. E

The spin faq of the principal axes of strain is related to the rigid body spin mU via the following equation [41

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312

~o - 0

a~ -

~

(&)~ + (&Y d ¢~ (~/)~-(%)~

q)-

where A~ are the principal stretches and

represents components of the rate of deformation tensor d u measured on the principal axes of strain. The rotation tensor R~..~ turns the spatial co-nrdinates x~ into the principal directions of strain. Replacing the elastic part in the Prandtl-Reuss equation [8] with the hypoelastic model proposed by Metzger and Dubey [4] mid the Poisson's ratio with a generalized material function

for

~=v

o

-(v

_< %

-v)ex~l

~,-~-~]Olf o r ~ >

ot

14)

gives, after further modifications [7], the Eulerian rate of stress

E

E

E

.

3E-2H/(1+~)

rI n 0

~

_

nrl ~-~ _ ( n - 1 ) , q

n

for

q > 0

where q

v=v

for

(2)

(5)

P

In equation (4) v is the Poisson's ratio for an elastic state, Vm,~ represents the maximum value of tile Poisson's ratio for a developed plastic flow, o~ is the proportionality limit and k is a scaling factor ranging from 1 to 10. In equation (5) E is the Young's modulus, ~J,, represents the flow stress, H / the

- (sud,~)l(%-~__

)

! / ?

is used to define whether a material del~rmcd ~s m a state of loading or unloading. Note that q is negative for unloading, in the case r~f ciastoplastic loading q ranges from 0 to 1. Tile materiai behaviour described by equation (5) can be divided int(~ three categories: the linear elastic, ~on-associaled elastoptastic and a~ssociated elastoplastic flow The linear elasticity is controlled by two standard material parameters, the Young's moduh/s E :rod lt~e Poisson's ratio v. Its limit is defined by li~c effective stress equal to the proportionality limi~ "~3nc nonassociated plastic flow is in cffecl tbr effective stresses greater than o~ and lower tha~ % The material's non-linearity is controlled by ~ posmvc odd number n. Once the effective stress reaches the yield stress the associated elastoplastic flow begins, assumed to obey the Huber-Mises yield ;:rtterio~ ~, 3. W E A K F O R M U L A T I O N "lt~e drawing process of an isotropic, ci astopiastic material, through the rigid die is considered. The following assumptions are made: i) the ~,c~rkpiece ~s drawn with a constant velocity v,, withoul back tension; it) tile deformation (~f matcria i~ axisymmetric; iii) The shear stress L ~m fl~e workpiece/die interface S is governed b 5 lhe, friction model

where m is the friction factor. The boundary value problem is tbrmulated as follows

a%/ox~ - o in

v

(':)',

elastoplastic hardening modulus, ~ the effective rate of deformation and s~j is the deviator of the Cauchy stress tensor c~j. A non-dimeusional quantity

~ op

in

V

ii0 t

313

o oo = oo

x-O

on

and

(11)

St

o

on

-0

Sf

(12)

V - Vo

on

Sv

(13)

vn - 0

on

S

(14)

where, % ~ and G, are shear stress, effective stress and normal stress, respectively; v. is the velocity normal to the workpiece/die interface; Sr represents a free surface; S, is a boundary surface on which the velocity v o is specified and S, is a boundary surface on which the stresses %o are prescribed. The final problem is to find the velocity field v~ and the Cauchy stress field crU which minimize the deformation power and satisfy equations (9) to (14). The solution is divided into two parts. The velocity field is computed first by minimizing the functional

ensure that the equilibrium and boundary conditions are satisfied. Therefore, as have been shown by the author and Lenard [7], the resulting stress field is not satisfactory. Another approach [7] leads to a system of partial differential equations governing the stress transport in steady state metal forming processes. The Cauchy stress field can be determined by minimizing the error norm

1 3

- o¢E - fla~ o~ + o~, Qq~ ] ~dV

+ f s v i O ~ n ~ ~S = W ~

fs ~UAvlldS +

fJs

• IA~41d S

(16)

under the constraints (9), (10) and (12), subjected to the initial conditions (11). It has been found, however, that additional constraints are necessary to stabilize the solution. For drawing processes they are proposed as follows

f$ vi°.n~dSv v "J J

6(1-2v)

3

~(TO- ~ ~ fv[(°°d°x')vk

fs'~,lAvldS

(17)

(18)

]

(15)

The multiplier 7 equals 1 for loading and -1 for unloading. For the first iteration the effective stress is set to op and the mean stress o,, to zero. In subsequent iterations ~ and c r are updated from the solution for the stress field. Therefore, the first part of the solution involves searching for the velocity field which makes the functional (15) stationary and satisfies equations (13) and (14). The second part of the solution involves the integration of equation (1) using the velocity field determined by minimizing the functional (15). The simplest way to accomplish it is to integrate equation (1) along streamlines which leads to a system of ordinary differential equations. This method is very efficient. However, it does not

The constraint (17) maintains the equilibrium between the power dissipated by external and internal forces # . Equation (18) imposes traction, caused by friction on the workpiece/die interface, on the stress field. Note that the terms on the right-hand side of equations (17) and (18) are known from the solution of the functional (15) for the velocity field. 4. R E S U L T S

AND DISCUSSION

The predictive capabilities of the proposed steady state formulation are demonstrated by a numerical test performed for drawing of a copper bar. The geometry of the drawn material shown in Figure 1 is discretized using a mesh of 8,35, 4-node elements, in the r and z direction, respectively. For the velocity and stress fields a mesh of 4 , 2 0 elements with parabolic and Hermitian shape

314 /'unctions, respectively, are employed in finite element approximations. The die is modeled by a rigid surface S. The process parameters are taken as follows: initial bar diameter of 27 m m , final bar diameter of 21 m m , die inlet angle of 32 degrees, die radius of 15 m m and the friction factor on the workpiece/die interface of m=0.1.

comparatively high due to the lriction p~cscribcd. 1: is observed that unloading at the exi~ leads to ~ significant longitudinal and circumtercn:ial tensile stresses at the bar s u r f a c e

400~--....................................... i =__::=

~z°3

F~-e

s,

s

200

l l

O'rr

' e

_. _. ,_ :. =

'"

i,r

0_tO

10

~0

80

40

~0

Z,

~0

ITI ITI

Figure I. The die geometry and the deformed finite element m e s h used in modelling of a bar drawing process.

The m o v e m e n t of the bar is enforced by setting the velocity v~=l m m / s on S,. The material parameters are assumed to be: E = 129600 MPa, n=3, v=0.343. 5=-4 and oh=60 MPa. The flow stress for copper is taken from [10] to be of the form o,, = 6 0 + 2 8 1 ( ~ n ) ° ' ~

3 e2(l+v)

(20~

30 mm

40

50

200T~ ......................................... ~r=~---,-,::

~o IOO~

r.o

i

,l

U3-100 i ~-~,..D-p-[] ::::o ,

The distribution of the Cauchy stress tensor components at the bar surface are shown in Figures 2 and 3. At the bar surface the radial stresses are compressive with the highest magnitude some distance from the entrance to the deformation zone. The circumferential and longitudinal stresses are tensile prior to the entrance and after the exit from the conical part of the die. Along the conical part of the workpiece/die interface o0o stresses are compressive due to the high hydrostatic pressure acting there as shown in Figure 3. Shear stresses c~,.~ at the workpiece/die interface shown in Figure 3 ~ e

..... 2'lJ z,

Figure ~, v Distribution of radial, circumlcrentia! ant! longitudinal components of the Cauchy ~:rcss tcns~,r at the surface of the drawn copper b:u.

( 1 ~) )

where the inelastic effective strain is related to the elastoplastic one via the formula

ein

-400~1.0

Ora ~r~

30

Z,

40

50

rnlT1

Figure 3. Distribution of the mean stress and the shear component of the Cauchy stress at the surface of the drawn copper bar.

The stress distribution in the bar centre are shown m Figures 4 and 5. The longitudinal stresses are iensile at the entrance to the die and increase along the

315

particle path. After reaching a maximum value near the exit from the conical part of the die c = stresses decrease slightly due to unloading and stabilize at a steady state value.

die and reaches positive values in the plastic zone. This effect is probably responsible for cuppy fracture observed in drawing.

300'

co&:'-

3O0 300

~

~L 1 0 0

~,0~71,C~ ~iYress~c1"e

c,,ccc=

250

800

15o

0

100

-100

"~ 5O

50

~-200 0

-300

........

-50

-400

......

i'o .......

z'6 ....... Z,

3'6 .......

~t'6 .......

5'o'"

i .........

10

i .........

20

Z,

i .........

30 mm

i .........

40

50

m m

Figure 4. Distribution of radial, circumferential and longitudinal components of the Cauchy stress tensor in centre of the drawn copper bar.

Figure 6. Distribution of the normal pressure and the shear stress at the surface of the drawn copper bar.

300: 2OO 100

too 0

~)

0

5Q -i00

~-100

-200 50-200

........~ .........4 .........8 .........g ........ i'o .......iz

r, m m coo.

-300

co

{Tll

. . . . . . . i ~ . . . . . . . z ' 6 . . . . . . . 3'6 . . . . . . . 4 ' 6 . . . . . . . 5 ' 6 ' " Z,

m m

Figure 5. Distribution of the mean stress and the shear component of the Cauchy stress tensor in the centre of the drawn copper bar.

It can be seen from Figure 5 that the hydrostatic pressure decreases rapidly after the entrance to the

Figure 7. The drawing stress p,, and distribution of the Cauchy stress tensor components along the bar radius after exit of the die.

The variation of normal pressure and the shear stress along the material/die interface are presented in Figure 6. The pressure distribution curve is characteristic for a bar drawing process, showing the maxima near entrance and exit from the die. The

316

shear stress variation follows a similar pattern. Figure 7 shows the radial distribution of longitudinal, tangential, radial and shear components of the Cauchy stress tensor after the die. The dashed line indicates the drawing stress. The residual stresses are compressive at the axis of symmetry, while at the surface the tangential and longitudinal stresses are tensile. The magnitude of radial and the longitudinal stress components is significantly lower than the flow stress. Note that no axial force, no torque, are transmitted by the steady state residual stresses and the stress tensor components are in equilibrium.

material seem to be responsible lk)r ~:tJl)py ffactu,~ observed in drawing processes. ACKNOWLEDGEMENTS

Financial assistance of the Faculty ~)l Meta}lurgy and Materials Engineering, University ~)t Mining and Metallurgy is _gratefully acknowledged Grant N~. I0.110.54 REFERENCES

1

5. CONCLUSION

The presented solution shows that the elastoplastic steady state formulation can be applied successfully to compute the stress field in a drawn material. The proposed finite element scheme for stress computation allows to satisfy the equilibrium equations directly at element nodes. The yield criterion introduced as an inequality constraint enables to avoid arbitrary choices concerning the yielded points. Formulation of the problem in the Eulerian reference frame enables the boundary conditions to be satisfied globally in the final configuration. The predicted tensile mean stresses acting in the plastic zone in the centre of a drawn

C.C. Chen, S.I. Oh, S. Kobayashi. T!a~.is. ASME J. Eng. Ind., I01 (1978) 23. 2. R.N. Dubey, SM Archives, I2 (i987) 233. 3. R.N. Dubey, SM Archives, i0 (1985) 245 4. D.R Metzger, R.N. Dubey, Int J Plasiicity. 4 (1987) 341. 5. R. Rubinstain, S.N. Atluri, Comp. Meth App]. Mech. Eng., 36 (1983) 277 6. J.L. Chenot, Eng. Computations, 5 t1988} 2. 7. Z. Malinowski, J.G. Lenard, Comp. Meth. Appl Mech. Eng., 104 (1993) 1. 8. K Washizu, Variational Methods in Elasticity and Plasticity, Pergamon Press, ()xlorc!. 1975. 9. Z. Malinowski, R. Szyndler. Stcei ~'csear~:]~ 58 (1987) 503. 1(1. L. Sadok. M. Padko, Steal reseaJci~ r:() :[989) 351