H Y D R O S T A T I C E X T R U S I O N O F R O U N D A N D S H A P E D T U B E S
TOMIHARU MATSUSHITA, YOSHIHIRO YAMAGUCHI, MASATAKA NOGUCHI and MASAO NISHIHARA
Central Research Laboratory, Kobe Steel, Ltd.,3-18, 1-chome, Fukiai-ku, Kobe (Japan) (Received July 11, 1977)
I n d u s t r i a l S u m m a r y
The manufacture of tubes of round and of complicated shape by cold or hot hydro- static extrusion is investigated with regard to the extrusion pressure, the tool arrangements and the productivi ty attainable with hydrostat ic extrusion.
The pressures required to extrude round tubes are 3 to 20% higher than those required for rods of aluminium, aluminium alloy, copper, carbon steel, zircaloy and stainless steel. For shaped tubes having complicated profiles, the extrusion pressures are 10 to 40% higher than those for rods.
Three kinds of tool arrangement for tube extrusion are proposed for increasing the product ion rate and reducing manufacturing costs. They use, respectively, a mandrel fixed to the connecting cylinder, a travelling mandrel not supported by the billet end and a floating-tipped mandrel in which the floating tip behaves similarly to the floating plug in tube drawing. By employing these tool arrangements, disadvantages of the conventional fixed and travelling mandrels can be eliminated.
The assessment of the product ion rate of copper tube that could possibly be attained by using hot hydrostat ic extrusion is also compared with that for the conventional tube manufacturing process. The opinion is advanced that a large number of drawing and tube reducing passes could be eliminated by applying hot hydrostat ic extrusion. Eccentricities of tubes as extruded range from 2 to 3%.
N o t a t i o n
R
P Pe P h P r P f P t P t Oy ~y
(~cy f/au T
e x t r u s i o n r a t i o l i q u i d p r e s s u r e e x t r u s i o n p r e s s u r e i d e a l w o r k o f h o m o g e n e o u s d e f o r m a t i o n p e r u n i t v o l u m e r e d u n d a n t w o r k p e r u n i t v o l u m e f r i c t i o n loss p e r u n i t v o l u m e c o n t a c t p r e s s u r e o v e r t h e t o o l a n d m a t e r i a l i n t e r f a c e a v e r a g e c o n t a c t p r e s s u r e c u r r e n t f l o w s t ress o f m a t e r i a l a v e r a g e f l o w s t ress o f m a t e r i a l c o m p r e s s i v e y i e l d s t r ess o f b i l l e t m a t e r i a l a u g m e n t i n g s t r e s s in c o n v e n t i o n a l t r a v e l l i n g m a n d r e l s h e a r f l o w s t ress
34
c
V S u
U*
v
rn
Ro, Ri ro, ri rc
rp rm t Lp Ld a
F1, F2, fl, f2
strain volume of deformed material area velocity of billet velocity discontinuity velocity of deformed material over tool surface shear factor average coefficient of friction semi-cone angle of die semi-cone angle of tapered tip of mandrel outer and inner radii of billet, respectively outer and inner radii of product, respectively radius of initial contact of mandrel tip and product outer radius of mandrel tip radius of stepped-end of mandrel wall thickness of product bearing length of mandrel tip bearing length of die radius of intersecting circle of inscribed cone of die and circum- scribed cone of mandrel tip forces acting on mandrel tip
1. Introduct ion
During the past ten years, particular attention has been focussed on the application of the hydrostatic extrusion process to the metal-working indus° try because of its superior lubrication, favourable stress conditions for metal deformation and excellent versatility. This process has been used already in some limited industrial fields, such as the manufacturing of copper-clad aluminium conductors and wires of precious metals. Recently, much effort has been made to apply this process to the manufacturing of high-quality seamless tubes of industrial materials.
Pioneering work on the hydrostatic extrusion of tubes was carried out by Beresnev [1] and Pugh [2] who used a fixed mandrel of which the rear end was attached to a support cylinder. Fiorentino [3] introduced a travelling mandrel of which the rear end was supported directly by the billet. Thompson [4] extensively studied the features of the augmented hydrostatic extrusion of tubes.
The first object of the present work is to show the dependence of the extrusion pressure for the extrusion of tubes on the billet material, extrusion ratio, type of mandrel and sectional shape of the product. The second object is to discuss the roles of tool arrangement in the extrusion of tubes. The third is to s tudy the productivity attainable with the hydrostatic extrusion process when applied to the manufacture of tubes.
35
2. Experimental procedure
A 5-MN horizontal hydrostat ic extrusion press was used. This press has two aligned containers. The bore diameter of the billet container is 48 mm and that of the pressure container is 60 mm. The maximum working pressure available is 1500 MN/m 2, and the maximum ram speed is 20 mm/s.
Five types of mandrel were tested in the present study (see Fig. 1). Figure l (a) shows a conventional fixed mandrel and Fig. l (b) a conventional travel- ling mandrel. Figures l (c) , (d) and (e} are three types of newly-introduced mandrel. The first is a fixed mandrel at tached to the connecting cylinder in the tandem container press {Fig. 1(c)). This is suitable for extruding a long billet of material having a high flow stress, such as stainless steel and zircaloy. A modified travelling mandrel, indicated in Fig. l (d) , is the second type. To prevent buckling and barrelling of the billet, the rear end of the mandrel slides over the bore of the container. The third type is a floating-tipped mandrel, which is applicable to any type of fixed mandrel. The tip is easily replaced.
The materials extruded were 1100 aluminium, 2024 aluminium alloy, commercially pure copper, 0.15% carbon steel, zircaloy-2 and 304 stainless steel. All hollow billets were fully annealed, and then extruded at room temperature using a mixture of castor oil and methanol. In addition, ho t hydrostat ic extrusion of commercially pure copper was carried out. In this case, the modified travelling mandrel shown in Fig. l (d) was used. After charging a lump of grease into the container, the preheated billet was loaded and extruded.
(a) (c)
(e)
Fig. 1. Five types of tool arrangement tested, employing: (a) conventional fixed mandrel; (b) mandrel fixed to connecting cylinder; (c) conventional travelling mandrel; (d) travelling mandrel without augmenting axial force on billet; and (e) floating-tipped mandrel.
36
3. Extrusion pressure for tubes
The extrusion pressures required to extrude round tubes from the six materials in the cold condition using the fixed mandrel are summarized in Fig. 2. A linear relationship is observed in tube extrusion between the logarithm of the extrusion ratio, R, and the extrusion pressure, Pe. The extrusion pressure for tubes is from 3 to 20% higher than that for rods. From the experimental data, the following empirical formulae were obtained:
P e / l n R = 57.9Hv + 660, for rods (1)
and
P e / l n R = 7 6 . 0 H v - 200, for tubes (2)
where H v denotes the Vickers hardness number of the billet material. Test results for the hot hydrostatic extrusion of commercially pure copper
are shown in Fig. 3. The temperatures in Fig. 3 represent the preheating tem- peratures of the billet, and the temperature drop was kept within 50°C during extrusion. In the cold condition, the maximum extrusion ratio employed was 30 with a pressure of 1500 MN/m 2. In the hot condition, however, the maxi- mum ratio employed was 100 at 400°C and 900 at 650°C with a pressure of 1200 MN/m 2. Copper tubes extruded in the hot condition are shown in Fig. 4.
% z
c l
k
u~
uJ
1 5 0 C 3 0 4 ~ ' I / c o m m e r c i a y . . . . . . .
Fig. 2. Observed ex t ru s ion pressures for r o u n d t ube at r o o m t empe ra tu r e .
37
1500
z 1000
~ 5 0 O
Z x kJ
RoOm temp 300 °C 400 °C
o I 10 100 1C00
Extrusion rat io, R
Fig. 3. Observed extrusion pressure for round copper tube at elevated temperature.
Fig. 4. Copper tube hydrostatically extruded in the hot condition, of dimensions: left, 10.8 mm OD × 0.4 mm thickness; centre, 11.8 mm OD × 0.9 mm thickness; and right, 12.8 mm OD x 1.4 mm thickness.
38
The theoretical and experimental extrusion pressures for tubes and rods of 1100 aluminium are compared in Fig. 5. For the calculation of extrusion pres- sure, eqns. (A16), (A17) and (A19) as given in the Appendix were used and the mean flow stress ~y was determined from
, / Ey = - - oyde,
e 0
where e -~ In R. Figure 5 shows that the opt imum included angle of die for tube is larger than that for rod for a given shear factor. The difference in opt imum die angle is due to the increased contact area between the mandrel and the billet and also due to the redundant work, which latter depends on the ratios of the inner and the outer diameters of the billet and the product .
£1
$ h 5 2 0~
egn.(A16)
4 - -
Calculated Experiment
Tube 0
Rod _ _ . . ~ •
Shear f a c t o r : m = 0.10 Ex t rus ion ra t i o : R = 13.5
~-~O
Avitzur's solution (egn. (A1B))
I I I I I 0 20 40 60 80 "100 120
Included angle of d ie ,2~
Fig. 5. Comparison of the o p t i m u m die angles for 1100 aluminium tube and rod.
Figure 6 shows the reduced excess pressure to extrude round tubes in rela- tion to the extrusion pressure for rods. The solid lines in Fig. 6 represent the reduced excess pressures calculated from eqns. (A16) and {A18), given in the Appendix, for tube and rod, respectively. The reduced excess pressures for difficult-to-lubricate materials that have large values of shear factor m are seen to be strongly dependent on the extrusion ratio (Fig. 6). The reduced excess pressures for tubes of an easy-to-lubricate material -- such as aluminium -- are less than 0.05 at high extrusion ratios. The calculated friction loss at low extrusion ratios for round tubes is twice that for rod extrusion.
Shaped tubes -- such as internally finned, externally finned, flat and hexagonal -- were extruded at room temperature. The reduced excess pres- sures for various shaped tubes are shown in Fig. 7. The excess pressure
39
3
uJ
0.3
0.2
O,1
f / / / ~ m=O.2
/o o ®%
® ®
O 1100 aluminium
~) 2024 aluminium alloy
commercially pure copper
(9 z i rcat loy- 2
(~ 304 stainless steel
- - C a l c u L a t e d by egn.(A16) and(A18) in appendix r i =Smm, ro=6mm, ~ = 2 2 . 5 °, (9=0 °
Shear factor m=O.1
@ 0
o I I 50 1OO
O I 10 500
Extrusion rat io, IR
Fig. 6. Observed and calculated reduced excess pressures for round tubes over the ex t rus ion pressures fo r rod.
0.5
o.~ P;
$ 8
steel 0 : ~ , 1 1 0 0 A I
0.2 -- ~ 2per
~ 1 ~ e x c e s s pressure for steel ~ / rnund tube
0.1 -- ~ , ~ -
c c~pDer ~'- ~ o ] I 1100AI I
5 10 5O 100
Extrusion ratio, R Fig. 7. Observed reduced excess pressure for shaped tube over extrusion pressure for rod.
requirement for shaped tubes is dependent on the cross-sectional shape of the product. This suggests that excess pressure is caused not only by friction loss but also by increased redundant work.
Typical tubular products are shown in Fig. 8.
40
Fig. 8. Typical tubular products.
4. Selection of tool arrangements for tube extrusion
A fixed mandrel held stationary by a supporting cylinder is commonly used in the hydrostatic extrusion of tubes. By using this type of mandrel, thin-walled and finned tubes can be extruded with good accuracy, but the method is not suitable for a long billet because the length of the latter should be less than half of the length of container to avoid blocking of the ram motion. In order to eliminate this difficulty, a tandem container press (Fig. l(c)) has been designed, which is suitable for extruding a long hollow billet of materials having a high flow stress, such as stainless steel and zircaloy.
The front end of a fixed mandrel can be provided with a selected angle of taper, and the mandrel itself then compressed in the axial direction, such that the compressive stress thus generated suppresses the tensile fracture of the mandrel. A floating-tipped mandrel has been designed by taking this stress condition into consideration.
Figure 9 shows the stresses acting on the floating-tipped mandrel. There are four force components f , , f2, F, and F2 that move the tip in the axial direction. Two of them, which are the frictional drag f, over the tip and material interface and the force f2 due to the liquid pressure acting on the surface S,, drive the tip toward the die exit. Assuming that the frictional stress m r is reduced to m ~ y / X / ~ , according to the yon Mises yield criterion, f, is given by
re /,, f , = f l 2rrr 'mT'dr / tan /~ + 2 n r i ' m r ' d L
ri 0
= nmf fy ( rc 2 - ri~)/vf3-tan~ + 2 ~ m r i f f y L p / x / ~ - (3)
rim-,
and/'2 is given by
f2 = ~ ( r p 2 - - rm2)Pe (4)
The two other force components , F~ and F2, repulse the tip. The force F~ due to the contact pressure over the tip and material interface is obtained if it is assumed, with reference to Kudo's work [8], that the average contact pressure/~t, is equal to mgy/(~x/3):
F1 = / c 2~Pt rdr = ~m°y(rc 2 _ ri2)/fix/-~- (5) ri
and the force F2 is given by
F2 = ~(rp 2 - rc2)Pe (6)
For the tip to be in self-equilibrium, it is required that
F1 + F2 = fl + f2 (7)
From eqns. (3), (4), (5), (6) and (7), the following is obtained:
l ( r c t 2 rm 2 !
m = - - (8)
l \ r i / ~ tan/~ -- 2 ri
4 1
$1
Pe
- - Pe
Fig. 9. Forces acting on the floating tip of the mandrel.
42
Figure 10 indicates the area of successful application of the floating- tipped mandrel as determined from eqn. (8). For the calculation, it was assumed that radius re in Fig. 9 was kept constant. If the average coefficient of friction is less than 0.073, which value is larger than that observed in cold hydrostatic extrusion [9], extrusion is possible over a wide range of pressures. The floating-tipped mandrel was used in the present tests in extruding alumi- nium tube at room temperature. The outer diameter of the extruded tube was 10 mm and the wall thicknesses were 1.5 mm and 2.5 mm. The semi- cone angle of the tapered tip was 20 °. Under these conditions, aluminium tubes were hydrostatically extruded at the same pressure as in hydrostatic extrusion with the conventional fixed mandrel.
1.0
0.8
E 0,6
0,4
0,2
r- c -- H-I- : 3.2 ~ ~
2~--= 2
_ /
I I I I 2 4 6 8
Reduced extrusion Dressure,~y
Fig. 10. Predic ted applicable cond i t ions for the f loat ing- t ipped mandrel .
The floating tip of the mandrel behaves similarly to the floating plug in the tube drawing operation. Replacing the tip is quite simple and the tip can be made of even a brittle tool material -- such as sintered carbide -- for in- creased tool life. This type of mandrel is useful for hot hydrostatic extrusion to produce thin-walled tubes from large billets.
When this type of mandrel is applied to commercial use, it is recommended that the tip is designed to float under the large friction between the tip and the material and to be fitted onto a reduced diameter portion of the mandrel bar, as shown in Fig. 9, because the tip motion will then be blocked at a predetermined position despite the variation of friction during extrusion. The dependence of the axial position occupied by the mandrel on the level of friction is clear from eqn. (8).
43
The commonly-used type is the conventional travelling mandrel, as shown in Fig. l (b) . The pressure requirement of this mandrel is reduced because of the augmenting force, which latter is p ropor t iona l to the liquid pressure. How- ever, when the wall thickness of the billet is small when compared with the mandrel diameter, the large augmenting stress causes buckling or barrelling of the billet. In order to prevent barrelling of the billet, the augmenting stress Oau must be smaller than the compressive flow stress of the billet material. This condit ion is expressed by the following inequality:
~(Prm 2) < Ocy (9)
Oau 7r(Ro2 - Ri2)
As the extrusion pressure Pe is equal to the liquid pressure p plus the augmenting stress, the following equation is obtained:
Pe = [1 + r m Z / ( R o 2 - Ri2)]p (10)
Substituting eqn. (10) into eqn. (9), one obtains
Pe
aau 1 + ~ m ) ( RO 2 - (R_~m)2~Ocy (11)
Figure 11 shows the augmenting stress calculated from eqn. (11) for 1100 aluminium tube. Extrusion pressure Pe was predicted by using eqn. (2). The augmenting stress is seen to be strongly dependent on the ratios of the wall thickness of the extruded tube to the radius of the stepped end of the mandrel, t / r m , and on the ratio of the outer radius of the billet to the radius of the stepped end of the mandrel, R o / r m . Where the compressive flow stress of 1100 aluminium is 60 MN/m 2, steady extrusion wi thout barrelling is feasible within the shaded region of Fig. 11. Thus the hydrostat ic extrusion of thin-walled tubes using the conventional travelling mandrel is possible under limited conditions.
As the newly-introduced travelling mandrel shown in Fig. l (d) was not directly connected to the rear end of the billet, neither barrelling nor buckling occurred even when tube extrusion was carried out under conditions of small values of t / r m and R o / r m .
The test results ment ioned above are related to a single-action press. For commercial use of hydrostat ic extrusion in the manufacturing of tubes, a double-action press is convenient, because the mandrel movement can then be controlled independently. The double-action press will enable the realiza- tion of all the features of the mandrel types newly-introduced in the present work. i.e., the use of the full length of the container, the employment of the floating-tipped mandrel and the elimination of buckling and barrelling of the billet.
44
160
14C
120
~E 10C I z E ; ~ 8O
d
L -~ 60
g 4o E
<
Ro
0.5
P
Ri = r i : r m = l O m m ro =12ram
Mater ial ' :1100 aluminium pc= (76.0 H v - - 2 0 0 ) I n R
O'cy
oi[ll I I I 1 I 1 2 3 4 5 6
Oute r rad ius o f bi l let Ro Radius of mandre l Pm
Fig. 11. Theoretical augmenting stress determined for the conventional travelling mandrel.
5. Productivi ty s tudy of tube manufacturing
5.1 Produc t ion rate The following is an assessment of the attainable productivi ty of thin-walled
copper tubes. As shown in Fig. 3, copper tube can be hydrostat ically ex- t ruded at an extrusion ratio of 500 and at a billet temperature of 650°C, where a pressure of 1100 MN/m 2 is required. Let us assume that a 35-MN double-action press is employed and that a tube of 20 mm outer diameter, 1 mm wall thickness and 620 m length is extruded from a 340-kg weight billet of 200 mm outer diameter, 50 mm inner diameter and 1.3 m length. This extrusion will be followed by three drawing passes to obtain the finished product of 10 mm outer diameter and 0.5 mm thickness. Assuming that the press is designed for 30 extrusions per hour and that the operat ion rate is 85%, 25 billets are actually extruded per hour. The yield of this process will not be less than 95%. The product ion rate in this case reaches 8075 kg per hour.
It has been repor ted that the manufacturing of copper tube by ho t hydrostat ic extrusion was successfully conducted in the Netherlands [9]. A press of 40-MN capacity with a fixed mandrel, as shown in Fig. l(a), was used. The pressure medium was castor oil and the billet was heated to 550°C. The ou tpu t of copper tube was around 2500 kg per hour, which is one-third of
45
that mentioned above; the increased output in the present assessment is mainly due to the raised billet temperature of 650°C and the use of the longer billet that is made possible by the employment of a double-action press.
In the most typical process of copper tube manufacturing, a parent tube is extruded from a billet, at a temperature of from 750 to 850°C and an extru- sion ratio of about 20, by a 25-MN press. To obtain a finished product of 10 mm OD and 0.5 mm thickness, one pass of the tube reducer and seven draw- ing passes are then needed.
Even in the most advanced hot ram extrusion [11], the maximum attain- able extrusion ratio is 99 with a 35-MN press, where the billet temperature is in the range 750 to 850°C. Tubes of 50.8 mm OD and 3.17 mm thickness and 44.3 m length are produced from a 218 kg weight billet, which tubes are then drawn to obtain the finished size, using a 2.13-m diameter horizontal bull-block; nine drawing passes are necessary.
The production rates of these three copper production processes are com- pared in Table 1. Judging from the figures in Table 1, the process based on hot hydrostatic extrusion is greatly simplified, although preparation of a hollow billet is needed.
T A B L E 1
Assessed at tainable p roduc t ion rates o f copper tube by ho t hydrosta t ic ext rus ion and convent iona l ho t ext rus ion
Hot hydros ta t ic Convent ional ho t Convent ional ho t ext rus ion ext rus ion (I) ext rus ion (II)
Billet weight 340 218 218 (kg)
Billet t empera ture 650 750--850 750--850 (°c)
Press force 30 35 25 (MN)
Tube reducer no t used no t used used
Drawing passes 3 9 7
Produc t ion rate 8.1 8.8 8.8 ( t /hour )
5.2 Concentricity o f product In the conventional tube manufacturing process that includes tube reducing,
eccentricity produced in the piercing operation prior to extrusion can be compensated during subsequent tube reducing and drawing. The eccentricity, as defined by (tmax - tmin)/(tmax + train) X 100% where tmax and train are the maximum and minimum wall thickness of the finished product, ranges
46
from 2.5 to 4 percent. On the other hand, in the conventional high-ratio extrusion process wi thout subsequent tube reducing, a small deviation of the mandrel from the centre can result in a great percentage variation in the wall thickness of the extruded tube because of the low wall thickness. A large eccentricity, once produced, cannot be eliminated in the cold tube drawing subsequent to the ho t extrusion.
Hydrostatically extruded copper tubes in the present tests had eccentrici- ties ranging from 2 to 3 percent. These values remained unchanged during tube drawing. Such superior concentr ici ty of the product was also observed in other experiments on hydrostat ic extrusion.
6. Conclusions
The pressures required to extrude round tubes of industrial materials are 3 to 10% higher than those for rod. However, round tubes of a hard-to- lubricate material -- such as stainless steel -- require pressures 10 to 20% higher than those for rod. For shaped tubes having a complicated profile, the extrusion pressures are 10 to 40% higher than those for rod.
Three tool arrangements for tube extrusion have been proposed for in- creasing product ion rates and reducing manufacturing costs. These use: a mandrel fixed to the connecting cylinder; a travelling mandrel not supported by the billet end; and a floating-tipped mandrel. These tool arrangements realize the use of the full length of the container, the employment of the floating-tipped mandrel and the elimination of buckling and barrelling of the billet.
An assessment of the productivi ty of the hot hydrostatic extrusion of thin- walled copper tubes has been discussed, and it is suggested that significant cost reduction could be brought about by adopting the ho t hydrostat ic extrusion process.
Acknowledgments
The authors would like to acknowledge many valuable discussions with Prof. H. Kudo of Yokohama National University and the help of Prof. Kudo in the preparation of the manuscript. They also wish to express appreciation of the helpful advice of Prof. T. J imma of The Tokyo Institute of Techno logy
References
1 B.L Beresnev, L.F. Vereshchagin, Yu.N. Ryabinin and L.D. Livshits, Some Problems of Large Plastic Deformation of Metals at High Pressures, Pergamon Press Ltd., London, 1963.
2 H.L1.D. Pugh, NEL Report No. 142 (1964). 3 R.J. Fiorentino, Proc. ASTME-CIRP Int. Conf. on Manufacturing Technology, A n n
5 H.S. Mehta, A.H. Shabaik and S. Kobayashi, J. Enng. Ind., Trans. ASME, 92(2) (1970) 403.
6 K.T. Chang and J.C. Choi, J. Enng. Ind., Trans. ASME, 94(4) (1972) 1108. 7 B. Avitzur, J. Enng. Ind., Trans. ASME, 86(4) (1964) 305. 8 H. Kudo, Int. J. Mech. Sci., 3 (1961) 91. 9 A.W. Duffill and P.B. Mellor, Annals CIRP, 17 (1969) 135.
10 Anon, The Engineer, July (1973) 32. 11 J.H. Cairns and D.B. Such, J. Inst. Metals, 98 (1970) 289. 12 B. Avitzur, Metal Forming; Processes and Analyses, McGraw-Hill, New York, 1967.
A p p e n d i x
Analysis o f the extrusion pressure for tubes The pressure for t u b e ex t rus ion t h r o u g h a converg ing die wi th a t ape red
mandre l is ana lyzed b y assuming t h a t the pressure is associa ted wi th {i) the w o r k of h o m o g e n e o u s d e f o r m a t i o n , (ii) the r e d u n d a n t w o r k and (iii) the f r ic t ion loss over the too l and mate r i a l in te r face re fe r red to un i t v o l u m e of the billet . The a s sumed k inema t i ca l ly admiss ible ve loc i ty field of Fig. 12 is e m p l o y e d for the c o m p u t a t i o n o f the ex t rus ion pressure .
r° L~
Fig. 12. Assumed pattern of deformation.
The f irst c o m p o n e n t o f the pressure r e q u i r e m e n t is de f ined by the w o r k of h o m o g e n e o u s d e f o r m a t i o n . With r e fe rence to Avi tzur ' s so lu t ion [ 12] , this is given b y
l n R
Ph = f(a,~) f ayde (A1) 0
48
where f(a,/3) is the func t ion re la ted to the semi-cone angles o f the mandre l t ip and die. When/3 < a < 65 °, f(a,~) -~ 1 wi thin a few percentage accuracy and eqn. (A1) can be a p p r o x i m a t e d by
R o 2 - R i 2 Ph -~ Oy In (A2)
ro 2 - ri 2
The second c o m p o n e n t refers to the r e d u n d a n t work. This depends no t on ly on the ge ome t ry of the too l bu t also on the assumed ve loc i ty discon- t inui ty . I t is assumed in this analysis t ha t the inner surface of the billet con- tacts the mandre l surface immedia te ly a f te r passing the d i scon t inu i ty line AB, as shown in Fig. 12. The work dissipated on crossing the d i scon t inu i ty line AB is given by
O
WAB = 2urdrzu* (A3) Ri
where u* is the ve loc i ty d i scont inu i ty , expressed by
u* - rRo tanc~ + rRi tang (A4) Ro 2 - Ri 2 r r
Integrat ing eqn. (A3), we have
2~ru W A s - 3(Ro: -- Ri2) { ( R o 4 - 3Ro2Ri 2 + 2Ri3Ro) tana +
+ ( R i 4 - - 3 R o 2 R i 2 + 2RiRo3)tanl3 } (A5)
The vo lume of material passing th rough the d i scon t inu i ty AB is
VAB = 7r (Ro 2 - - R i 2 ) u (A6)
The r e d u n d a n t work per un i t vo lume PrAB is t hen given by
The th i rd c o m p o n e n t is the power consumpt ion p f due to f r ic t ion over the die /mater ia l and mandre l /mater ia l interfaces.
The veloci ty v along the die face changes due to accelerat ion of material in the de fo rma t ion zone of Fig. 12. The veloci ty at a po in t E is
Ro 2 -- Ri : u (AIO)
v = (I t ans + a) 2 - (l tan~ + a) 2 cosa
The infini tesimal area of die face is
2n(l t ana + a) dS = dl ( A l l )
C O 8 ~
If cons tan t shear fr ict ion m r (= fly/X/-3) is assumed, the fr ict ion loss Pfd over the interface be tween the die and material is then,
1 1 2 l t ana + a Ro 2 -- Ri : 1
Pfd = u(Ro2 -- Ri2) Jl, cosa d l m r (l t ana + a) 2 -- (1 tanl3 + a) 2 cos~
2m~y [. tana (tan~ + tan~)(Ro - a) + 2a tana I
: x/-3cos2a(tans tan#) Ltan----an#s+t In - - - - ( tans + tan#)(ro - a) + 2a tans I
t ans + tanl3 + 2a t a n s / ( R o - a) ] -1--1n2 I ~ - + - ~ + 2a ~ : a-) (A12) J
Fol lowing a similar procedure for the interface be tween the material and mandrel , the fr ict ion loss Pfm becomes
2mSy [ t ana ( tana + tant3)(R o - a) + 2a tans]
Pfm = V~cos2~( t ans _ tan~) Ltans + tan~ In ( tans + ~ - a) + 2a tans , -
1 In t ans + tar~ + 2a t a n s / ( R o - a) 1 (A13)
J 2 t ans + tan~ + 2a t a n s / ( r o -- a)
Addi t ional power is consumed over the interface of the material and the bearing area of the die and of the material and the parallel por t ion of the mandrel tip:
_ 1 ( / . / a ) Pfdm lr(ro: -- ri 2) m r " 2 u r i d L + m r " 2nrodL
0
_ 2mSy (riLp + roLd) N/3-(ro: -- ri:)
(A14)
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Pe ~y
The extrusion pressure Pe is thus given by
1 - (Ph + PrAB + PrOD + Pfd + P f m + P f d m )
Oy
- 2 [t_Ro4-3Ro2Ri2+2RiaRo = In R°2 Ri2 + + ro 2 - ri 2 3 x / 3 - ( R o 2 - R i2 ) 2
+ r°4 -- 3r°2ri2 + 2ri3r° t t Ri4 - 3R°2Ri2 + 2RiR°3 (ro2 - ri2) 2 ~ tans + t . . . . (Ro 2 - Ri2) 2
+ri4--3ro2ri2+2riroa t ] 2m (ro 2 -- ri2) 2 i tang + x/~-(tana -- tan/3) ×
X [ t a n s + - 2 cos2a +--cos2/3 + tans + tan/3 \ cos:a cos2/3
2m + N/r~-(r02 - - ri 2) ( r i L p + r o L d )
where
+
(A15)
( tans + t a n / D ( R o - a) + 2a tans
A = In ( tans + tan/3)(ro - a) + 2a tans
and
( tans + tan/3) + 2a tan~/ (Ro - a) B = In ( tans + t an f l )+ 2a ~ -
Substituting/3 = 0 and a = R i = r i into eqn. (A15), one obtains the equat ion for calculating the pressure to ex t rude tubes using a straight mandrel :
+ro4--3ro2riE+_2ri3rot 2m t A B ( 1 )I (ro2 _ ri2) 2 ~ tans + X/r3-tana 2 co 2 + 1 +
2m X/~( ro 2 _ ri2) ( r i L p + roLd) (A16)
Subst i tu t ing a = r i = 0 into eqn. (A16), the fol lowing equat ion is obta ined for rod extrusion:
( m 2mLd Pe 1 + In + tans + - - (A17) ~y x / ~ s i n a cosa ~ r o ! 3 - ~ x / 3 r o
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Equat ion (A17) is similar to Avitzur's solution [7] if (4/3x/3)tana of eqn. (A17) is replaced by (2/x/3)[ (a/sin20~) - cot ~] which is derived from the assump- t ion of a spherical surface of velocity discontinuity. Avitzur's solut ion is e x pressed by
P e _ . _+ sin2 ~ cota + - - (A18) ~y x /~ tan~ ln~---~o, ~ - v ~ r o
