Specimen geometries predicted by computer model of high deformation forging

Int. Z Mech. Sci. Vol. 21. pp. 417-430 Pergamon Press Ltd., 1979. Printed in Great Britain

SPECIMEN COMPUTER

GEOMETRIES

MODEL

OF HIGH

PREDICTED

BY

DEFORMATION

FORGING

J. W. H. PRIcEr and J. M. ALEXANDER:~ Imperial College, London (Received 28 October 1978; in Revised form 26 January 1979)

Summary--This paper presents the specimen geometries predicted by a finite element model of isothermal forging. In isothermal forging the specimens are subjected to very high deformations and considerable flowof material results. The geometries are shown by a number of computer produced plots of finite element meshes which represent the flow of the material at various stages during the forging process. The predicted geometries are found to conform well with experiments for a variety of specimen and die configurations.

NOTATION

B m V Wp a ~ d~ do dA o,~ O0 ~X

a constant shear coefficient volume plastic work rate a large constant an element of the plastic strain sensor an element of the plastic strain rate sensor volumetric strain rate instantaneous viscosity an instantaneous constant an element of the deviatoric stress sensor equivalent flow stress shear stress a functional.

INTRODUCTION The work presented here was developed in conjunction with the experimental study of a process known as isothermal forming. In this process use is made of the fact that at high temperatures and low strain rates metals generally deform at low stress levels. In the process of isothermal forming the dies are heated to the same temperature as the specimen and the forging is carried out at low speed. This contrasts to normal hot forging where the dies are much colder than the specimen, the forging occurs at higher speeds, and the load is often applied several times using graduated dies. In isothermal forming the specimens can deform very extensively during the single loading and c o n f o r m closely with the shape of the dies. The dimensional accuracy of the product is very good and thus a minimum of further machining is required to obtain the finished product. The process has been used particularly for materials which are difficult to work and where scrap is undesired such as titanium alloys. Examples of the use of the process are given in a number of r e f e r e n c e s [ l ] and the experimental arrangements for the work described in this paper are discussed in another paper[2]. The experiments concentrated on titanium or steel alloys (of types

which resist oxidation at high temperatures) and the dies were made of IN 100. This paper concentrates on the diagrams produced by a computer program which was developed to model the experiments. These diagrams show the material undergoing very large deformations such as are obtained in isothermal forming, and it is found that the flow of the metal predicted by the program represents those seen in the process very closely. The basic modelling technique used the finite element method [3] to determine deformations and material properties were numerically interpolated from the data of Lee and Backoffen [4]. tNow at Nuclear Power Company (Risley) Ltd., Warrington. ~Now at Mechanical Engineering Department, University College, Swansea. 417

418

J.W.H.

PRICE and J. M. ALEXANDER

THE COMPUTER MODEL OF ISOTHERMAL FORMING Only an outline of the approach used for obtaining finite element solutions for large deformations is given here; a fuller description of the method used in this work is given in another paper [5]. Several special features of the isothermal forming of graphite lubricated titanium alloy specimens are included in this approach• (i) The material is strongly strain-rate sensitive but at the temperature of forming there is little strain hardening. The constitutive relationship must therefore be such that flow stress is a function of both strain rate and temperature but not necessarily of strain• (ii) There are high deformations obtained in the process and in most cases all the material is substantially deformed. The plastic strains that occur will be many times larger than any elastic strains. Hence a method in which the elastic strains are neglected will be satisfactory. (iii) The die-specimen interface normally exhibits no slipping with graphite, but some boundaries that originally are free eventually come into contact with the die by rolling onto the die surface. If lubrication such as molten glass is used then relative movement is observed. (iv) The dies can be considered to be rigid. From the consideration of these properties a finite element scheme was chosen that would give a satisfactory modelling of the process without too great use of computer time. (a) An increment of plastic work per unit volume, dWp, can be written, following Hill[6] as: dWp = o'~ide~i

(I)

where trljare the deviatoric stresses,,d~ are the increments of plastic strain,and the summation convention applies. This can be re-written.in terms of a plastic strain rate per unit volume. Wp = tr~:~j.

(2)

The normal elastic finite element formulation can be converted to use strain rates and deviatoric stresses instead of strains and stresses resp.ectively. The nodal variables are now considered to be velocities and the increments of deformation are really time steps. It is a work rate that is minimised throughout the volume. (b) For the constitutive relaiiol~ship the Levy-Mises flow rule can be used[6]. d~,~ = tr~idA

(3)

where dA is an instantaneous cotlStant. To facilitate calculations the equation must be written to involve time: ~,~= tr~i/2~.

(4)

It is written in this form in order to emphasise the connection between this formulation and that for the flow of viscous fluidsat very low Reynolds numbers. Thus v~ can be interpreted as an instantaneous viscosity.In the present program provision wa~ made for the "viscosity" to be estimated at each iterationfor each Gauss point in each element. The value was found from interpolation directly from experimentally determined stress-strain rate-temperature data (as is provided in Ref. [4]). The strain rate and temperature from the last iteration is used to estimate the viscosity for the current iteration. (c) Incompressibility must be ensured. Of the techniques which may be used to achieve incompressibility the most efficient and simplest is the penalty function approach[7]. In this approach a large positive number, a , is multiplied by the volume integral of the square of the volumetric strain rate: t~ f ~2 d V

(5)

and this is added to the work rate functional, X, to obtain the finite element virtual work rate formulation

x= f tC, dv + . f ~o2dV+ B

(6)

B represents the contributions of body forces (normally considered to be unimportant in slow speed plasticity problems) and surface traction rates. (d) The boundaries of the specimen are either free or geometrically determined by the dies if sticking friction, as described in the experiments above, is assumed. Frictional conditions have been allowed for in the present program. Provision is made for boundary nodes that were Originally free to roll onto the dies and become fixed. (e) During the deformation the mesh must be continuously updated to reduce the problem as far as it possible to one of infinitesimal strain. This achieved by using small time or displacement stet~s and each node is moved after each increment. The movement of each node is calculated by multiplying the velocity calculated at that node in the previous increment by the time elapsing between increments. The distorted elements and a high speed elimination solution for banded matrices. Technical details of this program are material has of its previous shape. As a result the mesh must be arbitrarily reformed only when it is absolutely necessary• This occurs at times when elements become excessively distorted or negative areas are created. (The original program was kindly provided by Prof. O. C. Zienkiewicz. It is an elastic program with isoparametric quadilateral elements and a high speed elimination solution for handed matrices. Technical details of this program are described by Zienkiewicz [3].)

Specimen geometries predicted by computer model of high deformation forging APPLICATION

419

OF THE PROGRAM

1. Cylindrical compression A test situation of a cylindrical compression test with no friction at the interface is a useful check on whether constant volume is obtained in the program. On carrying out such tests it was found that there was no detectable departure from constant volume, the mesh deforming uniformly and remaining always completely cylindrical. Such a test is shown on Fig. I. A more difficult situation occurs with sticking friction. Examples are shown on Fig. 2 of the deformation for cylinders of Ti6A 14V at 950°C, using material data from Lee and Backoffen [4]. The height to diameter ratios were originally ~, 1 and 4 respectively and as yet in none of these examples has any material rolled onto the dies. The most interesting example shown is the tall thin cylinder which exhibits a double bulge effect, a p h e n o m e n o n seen in experiments on specimens of this ratio. The deformation in the central region of the tall specimen is quite uniform because the effects as the friction at the ends becomes reduced. Fig. 3. illustrates a specimen originally of height to diameter ratio I which has been taken to a higher strain of 83% reduction. The free surface of this specimen has proceeded to roll onto the dies and also the mesh has been straightened out or reformed on one iteration because the movement of the nodes can cause the elements to cross one another in highly deformed regions. The original volume of this specimen was 14137 mm 3 but now it is approx. 13504 mm 3 measured from the diagram which is a loss of about 4.3%. The loss of volume is due to the fact that constant volume or zero volumetric strain rate is only met

o N

l-eo

T

O

Q O,

0'00

Z-00

4.00

6'00

8"00

10"00

IZ.00

4'00

6"00

8-00

10'00

IZ-00

,

:

4~00

6,00

8'00

~lr Z w

'9"

oo 0-00

Z'00

o

~o Z

~o o o.

,

0.00

2'00

I

10.00

12'00

FIG. 1. Compression of a cylinder without friction, one quadrant shown. Initial dimensions 25 mm high, 25 mm dia.

420

J.W.

I-I. P R I C E

and

J.

M.

ALEXANDER

o

o

to ¢g~

g d

i

0.00

4.00

8.00 CURRENT

i

17.00 16.00 MESH PROB 2

i

ZO.O0

!

24 O0

oo

8 tsa m

eb

------r"

•00

4.00

t.oo

t2.00

CURRENt MEStt PR0e 2 F1G. 2. Compression of cylinders of ratio ~, 1 and 4 with sticking friction. Half of each specimen is shown. exactly at certain points within each of the eight noded quadilateral elements. This is because these elements can only satisfy CO continuity across their boundaries, that is, only the velocities have been calculated with a continuous value throughout the body. The derivatives of the velocities are not continuous through the element boundaries and thus the strain rates, which are a function of the derivatives of the velocities must be discontinuous at the element edges. To achieve continuous derivatives, elements which enforce a C Ocontinuity are required, but such elements involve considerably more complication and many extra nodal variables. The trade-off in computer effort to achieve C' continuity is not justified in most problems (see Strang and Fix[8]). However, it is found that the derivatives of the nodal variables are accurately calculated at certain points within the elements and that the 2 x 2 Gauss points are normally very close to such points (see Naylor[9]). In the cases being investigated it is found that the volumetric strain rate is indeed very close to zero when calculated at the 2 x 2 Gauss points but finite values normally occur at the nodes. The values at the nodes can be either negative or positive and tend to cancel each other out. Figs. 3-5 illustrate high deformations of specimens of ratios 2/3, I and 3/2. Here the operation of the subroutines which enables nodes that are on the free surface at the beginning of the deformation to stick on to the dies is fully illustrated. The computing times for these high deformations is quite economical with the penalty function approach and these results (without stress calculations) were all obtained in less than two minutes of central processor time on the CDC 6400 at Imperial College. 2. Ring tests

Fig. 6 illustrates a ring test; the compression of an annulus between flat dies. The starting geometry has the ratio 4:2:1 (outer dia.:inner dia.:thickness) and the thickness of the specimen was 6.45mm. The frictional conditions are those of sticking friction and could be represented by a value of shear coefficient of unity, i.e. m equals one in the equation ~0 = m -:.

V~

(7)

where ¢ is the shear stress at the interface and c~0 is the equivalent flow stress of the deforming material. The factor of ~/3 is a consequence of the Von Mises flow criterion. The value of 60 is not a constant and is dependant on strain rate and temperature. Fig. 7 illustrates a similarly shaped ring, but in this case the friction subroutine was introduced into the program and a value of 0.06 was used for the shear coefficient. Fig. 8 illustrates a plot of reduction in inner

0'00

~G.

) 0,00

r

4"00

8-00

8"00

8.00

8,

800

12-00

12-00

12-00

2~oo

16.00

16.00

16-00

16'-oo

2()'00

ZO'O0

20-00

2().oo

I

4'00

8-00

\

12"00

\

4.ho

\ \

e:oo

I 0.00

%.~o

o-oo

!

,i.oo

4.bo

4.00

~ 4.bo

s.oo

,i-oo

12.00

,c;.oo

16.00

~, ~ ~ ~, ~ ~:oo 12'.oo ~i.oo

I1-00

20-00

2~.oo

20-00

~, ~ I 2(~-oo

S..

16'.00 26.00

-\ \ x , " , , > ~ L ' ~

i_ \ \ \\\~"~

°0~0

m

16.00

\ _ _ ~

i_ \ \ \\\\\'-~ \\\,~

0 ' _ 30

\

I ,I I ,/ / ,I / I,

~

I \ \ \ \ \ \ ~

I. I

I\ \ \ \ \ \ ~

~1 I !

'l

i t

2;.oo

24.00

3. Compression of a cylinder of starting height to diameter ratio 2/3 with sticking friction. Final compression shown is 82.5%.

~0

4-00

4.00

o

0.~

4.00

400

c' 0 O0

-1

o

ittl I

,,t q

:3

b)

3

:r"

~L

L

3

3

3

<

=,.

3

-i

3

)0 :1 'b

7"

£ [

)0

0-~

0'00

;g. t,;.

r~ r

.a

¢ir

£

_,2

2 "00

2"00

2"00 II

I0 "00

IO.O0 II

1

IZ-O0

12'00

12"0~

0"00

0.00

Z'O0

-'

2-00

\

\

4.00

6.00

8'00

" " "00 CURRENT MESH PROB

-

4.00 6.00 8.00 CURRENT MESH PROS

I0.00

I0'00 II

I0-00 II

II

IZ.O0

12"00

12 '00

;'IG. 4. Starting height to diameter ratio of 1 ; sticking friction. 54% compresssion shown in last figure.

4.1111 6"00 8-00 CURRENT MESH PROB

4.00 6-00 8.00 I0.00 CURRENT MESH PROB U

4,00 6.00 8"00 CURRENT MESH PROB

CURRENT MESH PROR

-n ),<

b.

r~

z:

Specimen geometries predicted by computer model of high deformation forging

g ±.

wo

,oz /

~g.

C~'O0 2'00 4"00 6"00 8"00 CURRENT MESH PROB II

10"00

/

°

~.-

0"00 2-00 4.00 6.00 S'O0 IO'O0 12"00 CURRENT MESH PROS II

~.~.

~

,

DO Z'O0 4"00 6'00 ~'00 10'00 12"00 CURRENT MESH PROS II

0"00 2"00 4-00 6-00 g.00 I0-00 12-00 CURRENT MESH PROS II

g °0700 2.00 4.00 6.00 8.00 10.00

°o.~

CURRENT MESH PROS II

I

~o

z.00 4.'~

•

8.~

~o.oo 12.oo

CURRENT MESH PROB II

\ \

Ig

0.00 2.oo 4~o °C

2.00 4.00 6.00 s.oo Io00 CURRENT MESH PROS II

g

,.~o ~.'oo t~.00 li.0o

CURRENT NESH PROS II

IJ.OO ~o.

6 ®

xo Z

//

~g D 0 ~00

, " 2.00 4:00 6:00 e:oo Io'.oo r2.00 CURRENT MESH PROB II

z.'oo 4.~o 6.00 s.'00 ,o'-00 t~.00 CURRENT MESH PROS II

.0

6' 0.00

2! 4o!6oo.ooooo

m

0.00 2'00 4-00 6.00 8.00 I0'00 12.00 CURRENT MESH PROB II

CURRENT MESH PROBE 11

FIG. 5. Starting height to diameter ratio of 312; sticking friction. Final compression shown is 60~. One quadrant only is shown.

423

424

J.W.H.

PRICE and J. M. ALEXANDER

!

oo e~

O

.o e~

8 O

g.l.,

.g

O

.m to

g !

!

O0 '1,

O0"Z

o

o

00"0

g

O t2

.g

6 ,q.

o a

00"9

w"

!

00'i, ooz 31~1/3H H,~SIX¥

00'0

o

6

8

I0

FIG. 7. Ring test with a shear coefficient is 0.06; half a specimen is shown. Final compression is 92%.

4

,,

mo

¢D

=r

o

O e~

B

'O

8

,.<

e~

ae~

_'d.

¢t

O

¢t -i

§,

O~

]

,

X

o....S

oj

/

I

/

/

/

o

//

X I

///X S ~ ~ . / X

I I

l

REDUCTION / IN HEIGHT

//

./1/ /

- - HAWKYARD & JOHNSON (70) "L UPPER CURVES FOR m - I - - - PRESENT ~E. RESULTS .~ LOWERCURVES FOR m-o.06

O UHLUBRICATEOEXPERIMENTS e GLASS LUBRICATED EXPERIMENTS

ZO "~m. 8. Comparison of Hawkyard and Johnson[lO] theoretical results, the present finite element predictions and some experiments for the ring tests.

~o

4o

60

/o REDUCTION IN BORE DIAMETER

0.-

),-

:lG. 9. Tangential strain rate distribution in the sticking friction ring test of Fig. 7.7 (first increment). Units are sec '.

0'

'0.

I0. I

a.

L

r"

Specimen geometries predicted by computer model of high deformation forging

Z~

~8 -------.e---

°0-00

4"00

8-00

IZ.O0

16"00

ZO'O0

FIG. lO(a)

427

428

J.W.H.

PRICE and :J. M. ALEXANDER

! P~6. lOCo) 1~o. 10. Compression of 17.5 mm hish, 17.5 mm dia. specimen with a profile die. The mesh is not reformed at any stage.

4.00

4-00

16-00 20-00 I

8"00 12-00 CURRENT MESH PROB 24 "00

0"00

0"00

4.00

4"00

8"00 ,2"00 16.00 2()'00 CURRENT MESH PROB I

,X

8-00 IZ-O0 16-00 20'00 CURRENT MESH PROB I

24.00

24-00

FIG. I 1. Compression with a profile die (see Fig. 10). The mesh in this case is reformed at increments 7, 12. The dashed line gives some close experimental results for the outside profile of comparable specimens.

0 "00

0"~

/ ,

=2

IZ'O0"

16'00.

Z4"O0

0.00

,

ZO'O0

ooo

4-00'

8-00"

z

l

8"00 IZ'O0 16"00 CURRENT MESH PROB I

I I~

.oo

4.00

12'00-

~,

16-00-

0"00

o.ii I

4~)0"

~ 8"00'

12"00"

,~

m. O

ga.

=r

o

O

E o

B

O

o"

g

=o

B

O

O~

430

J.W.H.

PRICE and J. M. ALEXANDER

bore diameter versus reduction in height and compares the results of some experiments, the current finite element results and the predictions of Hawkyard and Johnson[10]. The lubricant used for the low friction case in the experiments was molten glass, the high friction case used no lubricant. The mid-section radii have been used for all the comparisons (Hawkyard and Johnson assume plane sections remain plane and thus the mid-section radius for their analysis is the same as the end radii). Complete agreement with experiments cannot be expected because the real friction effects are not likely to be precisely the same as indicated in equation (7) throuthout the test, nor are the specimens accurately round after the compression. The net result suggests there is little to choose between the two theoretical methods of predicting ring behaviour. The neutral surface (the surface that does not move) can be determined for the sticking friction case using the hoop strain rate distribution from the finite element analysis. The neutral surface does appear to be very nearly cylindrical as is shown on Fig. 9.

3. Profile dies Fig. 10 illustrates the results for an extensive compression of a finite element model of the profile die experiments reported in Ref. [2]. Subroutines have been included in the program which allow not only the nodes from the outside surface to stick on to the dies, but also allow the nodes on the top surface of the specimen to stick to the conical surface of the top die. As will be noted in the latter stages of this compression, some of the element boundaries tend to cross each other and the die surface in the later stages of compression. Fig. 11 illustrates a modification of the~program which allows the mesh to be reformed at stages during the compression to remove this problem. Whenever the elements appear to be about to cross, in this modification the whole mesh is reformed in an arbitrary manner, but the exterior surfaces are not altered. The results of the finite element method are compared with experimental profiles that were measured from approximately similar specimens using a projection microscope. As can be seen the comparisons are quite satisfactory. The most important important indicator which occurs during this forming is the reversal of flow when material, instead o f flowing into the flanges from the bulk of the specimen, flows from the flanges into the conical region in later stages of compression. On the experimental specimens this is detected by the specimen increasing in height, a feature which can be seen to occur in the last stage of the forming. The finite element results on Fig. 11 are presented for a specimen that is as close as possible to experimental specimens. As can be seen, if the element boundaries joining the comer of the top die to the surface of the lower die are inspected the flow indeed does reverse in the finite element model and the height does increase in the last stage shown. CONCLUSIONS A finite element method has been described which can model the very high deformations seen in isothermal forging. The geometries shown in the various stages of the forging, and for a variety of specimens, are in good agreement with experimental results.

Acknowledgement--One of the authors U.W.H.P.) acknowledges the support of a scholarship from the Commonwealth Scientific and Industrial Research Organisation, Australia. REFERENCES 1. K. M. KULKARNI, J. Inst. Metals 100, 146-151 (1972); S. Z. F1GLIN, Kurznechno-Stanpovnochoe. 9, 6-9 (1972); F. W . BOI.X3ER, Forging Equipment Materials and Practices. U.S. National Technical Information' Service, MCIC-HB-03, pp. 213-245 (1973). 2. J. W. H. PRICE and J. M. ALEXANDER, 4th North American Metalworking Res. Conf., Columbus, OH, p. 46 (1976). 3. 0 . C. ZIENKIEWIEZ, The Finite Element Method in Engineering Science. McGraw-Hill, London (1971). 4. D. LEE and W. A. BACKOI~mN, Trans. A I M E 239, 1084 (1967). 5. J. M. ALEXANDERand J. W. H. PRICE, 17th Machine Tool Design and Research Conf. Macmillan, London (1977). 6. R. HILL, The Mathematical Theory of Plasticity, Clarendon Press, Oxford (1952). 7. O. C. ZIENKIEWICZand A. N. GODBOLE, Int. Syrup. on F.E.M. in Flow Problems, Swansea (!974). Wiley, 8. G. S1~.ANO and G. J. FIX, An Analysis of the Fit6te E l m e n t Method. Prentice Hall, N.J. (1973). 9. D. J. NAYLOR, Int. Z Num. Methods Engng 8, 443 (1974). 10. J. B. HAWKYARDand W. JOHNSON, Int. J. Mech. Sci. 9, 163 (1967).