The simulation of the viscoplastic forming process by the finite-element method

Journal d

ELSEVIER

Materials Processing Technology

Journal of Materials Processing Technology 55 (1995) 442-447

The simulation of the viscoplastic forming process by the finite-element method Mingwang Fu a, Z.J. Luo b aNanchang Institute of Aeronautical Technology, Nanchang 330034, China bNorthwestern Polytechnical University, Xian 710072, China Received 25 January 1995

Industrial summary

The rigid-viscoplastic FEM is used to analyze the viscoplastic forming process in this paper. The general rigid-viscoplastic FEM formulation is given and specific formulations of isothermal forging processes and superplastic forming are listed also. As an application, the combined extrusion process of pure lead, which is strain-rate sensitive at room temperature, is analyzed. The simulated results reveal the variation of the forming load with the stroke and its dependence on conditions. On the basis of the metal flow patterns defined by the rigid-viscoplastic finite-elementmethod, the change of the position of the neutral layer is given and it is found that the occurrence of folding at the flange may be attributed mainly to an abnormal flow pattern. Moreover, the calculated results bring to light the rule of deformation distribution and its dependence on strain rate.

1. Introduction

Viscoplastic forming processes can be defined as those in which the deformation of the material is influenced greatly by the strain rate. The forming processes of strain-rate sensitivity materials belong to viscoplastic forming, the isothermal forging process and superplastic forming being typical of this type of forming process. Because the flow of material is affected by the strain rate, the theoretical analysis of this process is very difficult by conventional methods and has become the main research field of plasticity forming theory. Cristescu is the research pioneer of the viscoplastic forming process, having studied viscoplastic flow through conical converging dies using a viscoplastic constitutive equation by means of limit analysis I-1-3]. The authors of the present paper have put forward a general method for the analysis of pressure forming of viscoplastic sheet metal under plane conditions by the slab method [4]; however, the calculated information given by this research method is limited. Comprehensive analysis and simulation of the viscoplastic forming process became possible only after the introduction of the viscoplastic finite-element method I-5-8]. In this paper, the general formulation of the rigid-viscoplastic method is given. At the same time, the specific formulations of the isothermal forging process and the superplastic forming process are listed, in 0924-0136/96/$15.00 © 1996 Elsevier Science S.A. All rights reserved SSDI 0 9 2 4 - 0 1 3 6 ( 9 5 ) 0 2 0 5 4 - J

light of their viscoplastic constitutive equations. As an application example, the forming process of pure lead, which is strain-rate sensitive at room temperature, is analyzed by the rigid-viscoplastic finite-element method. The calculated results reveal the nature of the variation rule of the forming load, the change of the position of the neutral layer, the flow patterns, distribution of the deformation and their dependence on the forming conditions.

2. Finite-element formulation

According to the rigid-viscoplastic variational principle, the actual solution gives the minimum value of the following functional:

O=fVoe(:)dV-fsFFUdS+fvo;=dv,

(1)

where: E(~) is the work function; SF is the surface on which the traction is prescribed; F is the traction vector; u is the velocity vector; 2 is the Lagrangean multiplier; and Vo is the volume of the deforming body. The work function E(~) can be defined as: E(~) =

f?

cri~d~i~ =

6 dg.

(2)

Supposing that the deforming body is divided into m elements connected by n nodes, the velocity vector in

M. Fu, ZJ. Luo / Journal of Materials Processing Technology 55 H995) 442-447

Eq. (1) can be determined by the following equation: U = NV,

(3)

where Vis the velocity vector at the nodal point and N is the velocity interpolation matrix. The strain-rate vector can be gived in the form:

= BU,

443

where: # is the equivalent stress; AV is the velocity increment vector; and it is the vector of the Lagrangean multiplier of each element (equal to the mean stress of the relevant element).

R=~

(6/~) 1_.al.q/,.TpT

(4)

where B is the strain rate-velocity matrix. The effective strain rate can be expressed as follows: = (~TD~) = ( v T p v ) 1/2,

(5)

where

C = { 1 1 1 0 0 O}T When the viscoplastic constitutive equation takes the form: e - - Y(g)[1 + (~/r)"]

2/3

then R becomes

0 2/3

R = {[(n 2/3

D=

1)Y(g)$"/r"-

(9)

Y(~)]/~:3}pwTpT.

(10)

In the superplastic forming process, the constitutive equation is the Beckofen equation:

1/3 1/3

ff = k~',

1/3

(ll)

so that R becomes:

P =

BTDB.

With discretization of functional (1), the functional ~bcan be approximated by:

~ ~ ~J(W,2J),

R= [k(m-

1)~'n/~ 3] P VI/TP T.

(12)

The actual solution of Eq. (8) can be obtained by the iteration method.

(6)

j=l

where: ~bj is the function of thejth element; and W and 2 s are the nodal velocity vector and the Lagrangen multiplier of the jth element. According to the stationary condition, the actual solution can be obtained from the following equations:

i =,

3. Viscoplastic forming material and experimental device In order to verify the results of analysis of the viscoplastic forming process by the FEM, the combined extrusion deforming process of viscoplastic forming is simulated, simulation material being pure lead, which has viscoplastic behavior at room temperature. The

(7)

j=l

/

Eqs. (7) can be linearized by the Newton-Raphson method, and the following linear equation system obtained:

[ ~ fv,,j(#/~)Pdv+fVo~Rd°

~ ~vosBTcdv 1

Plate Upper die

Middle die Bottom die

j=l

j=l f fSl~j [ it" J

~'J

Punch

_ ~ fVojVTBTcdv j=l

, (8)

J Fig. 1. Experimental device.

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M. Fu, ZJ. Luo /Journal o f Materials Processing Technology 55 (1995) 442-447

experimented device is shown in Fig. 1. The viscoplastic constitutive equation of pure lead is Eq. (9), where Y(g) is the static yield stress, ~ is the equivalent strain rate, r is the parameter associated with the viscosity of material, and n is the strain-rate index.

4.2. Friction condition The friction stress is determined by the following equation: r = mK[(2/n) tan(v~/a[vDf)]t

(17)

6.380 + 53.9g for g~< 0.143 Y(g) =

13.955 + 20.286 s i n [ ~ ( g - 0.143)/0.237] 15.758 - 2.685(g- 0.310) 14.543

1.485 - 6000g + 9.400g z - 4.600e 3 for 0.407 < g ~< 0.668 0.30

(]3)

for g ~< 0.407

(14)

for g > 0.668

- 14.140g + 6.426 rn

for 0.294 < g ~< 0.760

for g > 0.670

0.512g + 0.081

nf

for 0.143 < g ~< 0.294

for g ~< 0.384

10.882 - 52.182g + 86.718g 2 - 46.509g 3 for 0.384 < g ~< 0.621 0.781

The data in Eqs (13)-(15) is taken from [8].

where m is the friction factor (0 < m < 1), K is the shear yield stress, V, is the magnitude of relative velocity between the die and the workpiece, a is a constant several orders of magnitude smaller than the die velocity (such as 10-5), IVD[ is the absolute value of the die velocity, and t is the unit tangential vector.

4. The treatment of calculating technical problems 4.1. The position of the neutral layer The neutral position of the layer is defined as the place which is the watershed of metal flow. In order to determine the position of the neutral layer position, it is necessary to describe the side surface 345 of the quadrilateral element E with eight nodes, illustrated schematically in Fig. 2. Suppose that the side surface 345 is divided into n equal-length sections by (n - 1) equidistant points. At the nth step, if the velocity of equi-distant point i in the z direction satisfies the following equation: Vi n9 = N 3 i V ~ z +

(15)

for g > 0.621.

n + NsiV~z = 0, N4i V 4z

(16)

then point i is the neutral-layer point. The metal above this point flows upwards and that below this point flows downwards.

7

6

5

4.3. Mesh re-zoning Mesh re-zoning consists of generating a new mesh system and re-defining the field variables in the new mesh system on the basis of those in the old system, and can be accomplished by the use of an available algorithm. In this paper, authors take into account that the accuracy of calculation of the field variables at the Gauss integration point inside the element is much greater than that at the element boundary and that the field variables associated with the derivative of the velocity may be discontinuous at the element boundary. Thus the field variables at the element boundary are modified according to that at the inside Gauss point. Then the new field variables of the new mesh can be interpolated on the basis of the modified field variables in the old mesh.

i 8

E

1

4

2

Fig. 2. The determination of the position of the neutral layer.

5. Results and discussion In order to study the influence of deformation of strain rate and lubrication on the forming load, the distribution and the variation of the position of the neutral layer. The following forming conditions are used in the calculations (1) go = 10- 3 s- 1, m = 0.22 (~ is the initial strain rate);

M. Fu, Z.J. Luo / Journal of Materials Processing Technology 55 (1995) 442-447

(2) ~ o = 3 × 1 0 - 2 s -1, m =0.22; (3) ~ o = 3 X 1 0 - 2 s -1, m = 0.05. The calculated results and experimental results are as follows.

5. l. Forming load-stroke curve The forming loads for the three kinds of forming conditions are shown in Fig. 3. From the forming load stroke curve, it can be seen that: (1) the calculated loads are quite consistent with the experimental loads, the greatest difference being less than 8%; (2) the greater the strain rate, the greater the forming loads, i.e. the sensitivity of the material to strain rate makes the forming loads increase with the strain rate; (3) the forming loads increase with the stroke, because the strain rate and the degree of three-dimensional compressive stress are raised with the stroke; and (4) the frictional coefficient has a great influence on the forming loads.

445

5.2. The variation of the position of the neutral layer The flow patterns determined by FEM are illustrated in Fig. 4, where H represents the position of the neutral layer. On the basis of the calculated results, all metal flows downwards at the initial stage but when the punchstroke is 3.6 mm, the neutral layer appears: the metal above the neutral layer moves upwards and that below the neutral layer moves downwards. As the forming proceeds, the position of the neutral layer moves downwards. When the metal coming from the upper exit contacts the shoulder of the punch, all metal moves downwards and the neutral layer disappears. However, when the lower cavity is almost filled with metal, the neutral layer occurs again, but it is now located near to the lower exit. The variation of the position of the neutral layer with the stroke is shown in Fig. 5.

5.3. Deformation homegenity In order to investigate the influences of strain rate and lubrication conditions on the distribution of deformation. 30,

--

Calculated results Experimental data

A

•

o ~0"= 3 x 1 0 s ,m=0.22

/ '

"O

m 20 O E

'

=

-

-'

=

'

~o = 3 x 1 0 2 s

y

-1

m = 0.05 ~o = 10-3s 1

~ lO

h

~:o= 3 x 10-2s 1 m=0.22

m = 0.22

A E E v

E= .$

-r , 4

, 8

, 12

, 16

15

10

5

I. H

0

Stroke ( mm )

5

10

Fig. 3, The load-stroke curve.

H

Fig. 5. The variations of the position of the neutral point with stroke.

J I ,e q'~fSJ ,l p, f

H = 4.0 mm

15

Stroke ( mm )

H =6.0 mm

H = 16.0 mm

H = 14.0 mm

Fig. 4. The flow pattern determined by F E M (go = 10 3 s

1; m

=

H = 15.3 mm 0.22).

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M. Fu, Z.J. Luo /Journal of Materials Processing Technology 55 (1995) 442-447

o

H = 4.0 mm

I

H = 4.0 mm

H = 4.0 mm

H = 6.0 mm

H = 6.0 mm

H = 6.0 mm

H = 14.0 m r n

N = 14.0 mm

H = 14.0 mm

I

J N = 15.3 mm

H = 15.3 mm

( 1 ) ~ = 1 0 3 s -1,m=0.22

( 2 ) ~ = 3 x 1 0 s ,m=0.22

-3

1

H = 15.3 mm - 3 -1

( 3 ) ~ = 3 x 1 0 s ,m=0.05

Fig. 6. The distribution of 0 (MPa).

The three kinds of formation conditions are used in the F E M calculations, the results being given in Fig. 6. From this figure the following conclusions can be drawn: (a) The zone of greatest deformation of the workpiece is at corner of the punch bottom. (b) The zone of greatest deformation moves downwards gradually as the stroke increases. Moreover, the

gradient of the distribution of the deformation decreases and the deformation of homegeneity is improved with the stroke. (c) The deformation of homegeneity is improved with increase in the strain rate due to the sensitivity of the material to strain rate, and it is also improved with the improvement of the lubrication conditions.

M. Fu, ZJ. Luo /Journal of Materials Processing Technologv 55 (1995)442~147

6. Conclusions

(1) The general viscoplastic FEM formulation and the specific formulations of isothermal forging processes and superplastic forming given in this paper can be used in the analysis of practical viscoplastic forming. (2) The forming loads calculated by viscoplastic FEM are quite consistent with experimental values and can be used as a basis for choosing the forming equipment and determining the processes. (3) The flow patterns, the variation of the position of the neutral layer, and the distribution of deformation can be revealed by the simulation of the whole forming process with viscoplastic FEM. Their dependence of the forming conditions can be studied also. References [1] N. Cristescu, Plastic flow through conical converging dies using a viscoplastic constitutive equation, lnt J. Mech. Sci., 17 (1975) 425 433.

447

[2] N. Cristescu, Drawing through conical dies An analysis compared with experiments, Int. J. Mech. Sci., 18 (1976) 45-49. 1-3] N. Cristescu and S.I. Suliciu, Viscoplasticity, Martins Nijhoff, Dordrecht, 1982. I-4] Z.J. Luo and Guo Naicheng, A general method for analysis of pressure forming of viscoplastic sheet metal under plane strain condition, in: Proc. 15th NARMC, 1987, pp. 335-339. I-5] S.I. Oh, N. Rebelo and S. Kobayashi, Finite element formulation for the analysis of plastic deformation of rate sensitive materials in metal forming, in: I U T A M Syrup. on Metal Forming Plasticity, Tutzing, 1978, p. 273. 1-6] P. Hartley, C.E.N. Strugess and G.W. Rowe, Finite- element prediction of products, in: Proc. 8th NARMC, Rolla, M0, 1980, pp. 121-128. 1-7] S. Kobayashi, Thermoviscoplastic analysis of metal forming problem by finite- element method, in: J.F.T. Pattman, O.C. Zienkiewicz, R.D. Wood and J.M. Alexander (Eds.), Numerical Analysis ~]'Forrnin,q Processes, Wiley, New York, 1984, pp. 45-69. [8] Mingwang Fu and Z.J. Luo, The prodiction of macrodefects during the isothermal forging process by the rigid viscoplastic finite-element method, J. Mats. Proc. Tech., 32 (1992) 599 608.