rotary processes of metal forming, such as Malmesman piercing, ... Table 1. Initial data introduced into theoretical analysis of rotational compression. No.
Journalof
ELSEVIER
Journal of Materials Processing Technology 60 (1996) 549-554
Materials Processing Technology
Analysis of plane-strain rotational compression of rod by FEM Zb. Pater Technical University o f Lublin, Nadbystrzycka 36, 20-618 Lublin, P o l a n d
Abstract In this study, rotational compression of rods has been considered theoretically. Calculations were performed under the assumption of plane strain conditions, using commercial system FORM-2D based on the finite element method. The following topics have been considered: the width of contact area, the contact pressure and rotational compression stability. Calculated results have been compared with experimental data from current references. Keywords: Rotational compression, theoretical analysis, FEM.
Notations R Tbdo h ip rro AI Ar66E uo-o ~, ~ , Cry, o'~co o~, -
radius of roll or concave die temperature width of contact area initial diameter of rod distance between dies; h = r slipping coefficient contact pressure radius after rotational compression initial radius of rod displacement depth of rotational compression; Ar=ro-r relative reduction; 8=ro/r conventional strain strain rate effective strain friction coefficient yield stress mean stress normal stresses angular speed of workpiece angular speed of imaginary shaft
1. Introduction Rotational compression of rods is the basic method among rotary processes of metal forming, such as Malmesman piercing, cross wedge rolling and helical rolling. Proper design of the above mentioned methods of processing can be done if two characteristic features are known. 0924-0136/96/$15.00 © 1996 Elsevier Science S.A. All rights reserved PI10924-0136 (96) 02385-0
These are: the contact area between workpiece and die and pressure on the contact area. The majority of studies of rotational compression were aimed at specifying the magnitude of force acting on the dies during forming of the product. Because of the complexity of the considered problems, the initial considerations were limited to analysis of the static process of side-pressing of cylinders. First, solutions [1-2] should be mentioned, obtained using slip-line fields method, and the solutions [3j obtained using FEM. In many studies performed later rotational compression kinematics was taken into account. These studies started with determining the value of contact area width and the value of shear stress on that surface. Known solutions relate to various methods, namely: slip-line fields method [4-6], energy method [7] and upper bound method [8-10J. In this paper, a process of rotational compression of rods is studied and results of FEM-based analyses are presented. The processing is realised under assumption of a plane state of strain. The results of computations have been compared with experimental data available in references. 2. Rotational compression process modelling Modelling the rotational compression of rods has been performed using commercial computer code FORM-2D based on the finite element method. This computer system allows to simulate axially symmetric metal forming processes. It also enables carrying out simulation of processing with the plane state of strain. Calculations of metal flow scheme, strain, stress, strain rate and thermal fields are performed for the assumed rigid-viscoplastic model of material. In the analysis included were processes of rotational compression of rods made of: commercially pure aluminium,
Zb. Pater/Journal of Materials Processing Technology 60 (1996) 549-554
550
Table 1 Initial data introduced into theoretical analysis of rotational compression. No.
Process parameters
Yield stress of material
Dies geometry
After M. Hayama [11]:
A
for ~ < 0.3
material: lead friction coefficient: 0.1+0.5 relative reduction: 1.02+ 1.22 environment temperature: 20°C relative speed of two dies: 0.4 m/s
O-° = 8.84 (0.3 + 6. ~). ; °'' 27-°'22';
for ~ > 0.3 tro= 8.84 (1.65 + 1.5. ~ ) . ; 0'°67-°'0.9;
O16o !i!iii!iii!ii!!
After G. Ya. Goon [12]
material: alumininm friction coefficient: 0.3 relative reduction: 1.005+ 1.02 environment temperature: 20°C relative speed of two dies: 0.0055 m/s
~
E
140 -
2.5
ff 12o-
60
material: steel 45 (0.48% C; 0.53% Mn; 0.22% Si; 0.028% S; 0.025% P; 0.07% Cr; 0.11% Ni) friction coefficient: 0.5 relative reduction: 1.05+ 1.2 environment temperature: 20°C relative speed of two dies: 0.4 m/s initial temperature of the workpiece: 1000°C average temperature of dies: 200°C
t
100 - -
'1' 0.0 0.1 0.2 0.3 0.4 0.5
' ~ ' t ' 0.6 0.7 0.8 0.9
o t
,,'d"
---qIR12______6_6 2t
After G. Ya. Goon [12] 350 ~ ~ T l°Cl .... [ ........... i .......... i t tU,l 90o1100011100 1200 :: i i 300q 0.5 " 1 • I • * ---i ..........i...........i ] s.o ~1ol [] i i i 50.0 A I O I O O II! ..................... i
~
200
.................
150 100 ,
~
'
~
~
. . . . . . . . . . . . . . .
0.10
lead and grade 45 steel (according to GOST). Data used in calculations are specified in table 1. It should be noted here, that values of parameters of rotational compression were selected especially to make possible comparison of performed calculations with the published experimental data. FEM computations require partitioning of the continuum medium into a number of elements linked by a finite number of nodal points. System FORM-2D comprises fully automated mesh generator that divides a plane into curvilinear triangles, separately for each iteration step. An exemplary subdivision of
0.15
0.20
0.25
. . . . . . . . . . .
0.30
0.35
0.40
a product into elements, in cause of rotational compression, is shown in fig. 1.
3. Results
and
discussion
Below listed results of theoretical analysis of rotational compression of rods have been obtained under assumption of plane-strain conditions of processing.
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Zb. Pater/Journal of Materials Processing Technology 60 (1996) 549-554
o.so j
FEM
........... f.................. i................... i.................. i ............................ ..ii........ .• ....... ,~................. i • i i$q.(3) ! ...........r.............. • i ............... o-~--~............. ! I i •i ! /. [...........i.................. i...........~ - - - - ~ J *• • ~ i / i
~x=O.1 Ix=0.2 0 . 4 0 -q - I.).=0.3 0.45
,A- t3 -
4 0.35 q
o-~0.4 /x _ Ix=0.5
4--~
0.30
..................
i ...................
~ o . , - t .................. i
(a)
~ ~ E ~ ' -
q "
. :
~ i
i
3. I. Width o f contact area
...................
°-"l"Z/ 0.10 0.05 0.00
"-~"')~AI~........~
t//¢'*
--[f~
(1)
W. S. Smirnov et al. [14]
............. i .......... ..... *
i
i
........
".........
)
1.00
- v" Kasuga [17] M. Hayama [16] • - W.A. Kluszin [18] • - Zb. Pater [19]
•
f
[
Until recently, finding the width of contact area was the starting point of determining: contact pressure, forces, energetical parameters and conditions of stable processing. The width of contact surface was determined using, among others, dependencies of A. I. Tselikov [ 13]:
b = 2~ r~°
~
o.2o
(b)
Fig.1. Partition of workpiece into elements for rotational compression with 8=1.06, /.t=0.2: a) preliminary stage; b) advanced stage.
/2Ar(r+Ar) b =.i - - ~ ' VI+ ~-
-
-
t
I
I
1.05
1.10
1.15
i
I
1.20
1.25
8 Fig.2. Ratios b/do as functions of relative reduction 6, calculated using empirical formulas and FEM, and measured in rotational compression. agreement with b/do values calculated on a basis of the dependence (3) proposed by the author. 3.2. Contact pressure
(2/
and Zb. Pater and W. Werofiski [151 b = 3 _ ~ ro "
As mentioned above, former theoretical studies on rotational compression were limited to predicting the mean contact pressure only. Now, application of F E M enables, among others, predicting the distribution of contact pressure. In fig.3, contact pressure distributions obtained with FEM simulation of rotational compression are depicted together with 350
8 It should be noted that the condition R=oo is assumed in case of rotational compression with fiat dies. In case of roll, the ratio rolR - see dependencies (2) or (3) - is taken with the "+" sign, and in case of concave die with the "-" sign. F E M simulations of rotational compression allows not only to determine the width of contact surface, but to predict the shape of compressed specimen also. Values of the width b of contact area, obtained using dependencies (1)-(3), are specified in fig.2 together with the results of experimental of M. Hayama [16], Y. Kasuga [17], W.A. Kluszin [18] and own [191. Also in that figure dependencies are shown between b and friction coefficient/t and relative reduction ~ - predicted by FEM. The calculations have been performed using data A from table 1. From presented diagram it follows that the width b of contact area is essentially influenced by the relative reduction 6, and that b increases with the increase of 6. Greater values of width b are also related to greater values of friction coefficient p, but here the increase is not significant. However, a fact deserves notice that/a influences decisively on a maximal possible value of relative reduction 6 obtained in rotational compression. It also noticeable that values of b/do predicted by FEM are lower than experimentally obtained ones. Yet, they are in a quite good
FEM [ Exp.[20]
"''0100":':; 50 0
' 0.0
I 1.0
'
~
I 2.0
'~
--7 3.0
b [mml Fig.3. Distribution of contact pressure in rotational compression of aluminium bars.
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Zb. Pater/Journal of Materials Processing Technology 60 ('1996) 549-554
350
1.30
12,ill iiii[iiiiiiiiiiiiiili!!iiii iii iiiiiiiiiiiiiiiiiiii
i @ 6=i.oI •
300
•
•
--
6=1.05
-
6=1.10
-
6=1.15
25O 150
,
..................
i ................... i ................. i ................... i ............... •
00
1.1, .................. i................................... i ........ " ...................... i
m, 15o
I.I0 lOO 1.05 t .................
50
,
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
1.00 I 0.0
'
I 0.1
• - Experiment [18] '
0.2
I
0.3
'
I
0.4
'
0.5
b [mm] Fig.4. Distribution of contact pressure predicted by FEM for rotational compression of 45 grade steel bars.
Fig.5. Limiting relative reductions 8 that can be achieved in rotational compression under given conditions of friction.
experimental data measured by Y. Kasuga [20]. Parameters for FEM simulations were taken from line B, table 1. It can be seen that theoretical results well agree with the experimental ones. It follows from that figure tat contact pressure is not distributed uniformly across the width of the surface of contact. The maximum values appear near the inlet point. A similar characteristic of the distribution of contact pressure was obtained after rotational compression of 45 grade steel, using parameters listed in line C, table 1 - see rigA. Such findings, resulting from experimental work, have been also reported by G. W. Andrejew et al. [21].
In fig.5 the maximum values of relative reduction 8 are diagrammed versus p. These values were calculated from equation (5) and measured in tests [18]. Observed considerable discrepancy between the data first seems to be a result of differences between the actual and A.F. Lisoczkin's cross sectional area shape and distribution of contact pressure. Application of FEM to the analysis of slipping enabled the author more precisely determined values of the limiting reduction 6, see fig.5. It also made possible observation of a transition from stable to unstable state. From these observations it follows, that a transient phase appears between the stable and unstable processing phases. A characteristic feature of that intermediate phase is change in position of the instantaneous centre of rotation, from one die to another. An exemplary diagram of rotational compression, with stability limited due to slipping, is shown in fig.6. It has been also found out that the transient phase influences critically the possibility of determining the limiting relative reduction 6, see fig.5. Presence of this phase brings troubles because it is difficult to classify given process as stable or unstable. Another, quite frequently existent, limitations of stability in rotational compression are central cavities formed in processed products, known also as the Mannesman effect. According to G. Thompson and J.B. Haywkard [23], these cavities are fonned due to low-cycle fatigue of material. The mentioned authors have found out that in majority of events of rotational compression (when 5>1.01 according to W. Ya. Kluszin et al. [18]), material layers compressed in normal direction. Atter 1/4 of revolution, the compressed layers of forging are subject to stretching and the stretched ones are subject to compression. Such state of stress, after a number of cycles of strain (a double number of revolutions of the forging), can result in formulation of cavity in central zone of the product. Existence of such a state of stress in axial zone has been confirmed by results of calculations performed with FEM, using parameters A, table 1. The distribution of mean stress in rotational compression for ~5=1.05; 1.10; 1.15; and 1.20 is shown in fig.7. It can be seen
3.3. Rotational compression limitations
The process of rotational compression may become unstable, mainly due to slipping between the forging and the die. To give a definition of slipping, a notion of a coefficient of slipping i has been introduced. It is defined as a ratio of angular speed of workpiece co and angular speed co, of imaginary shaft, rotating with the tangential velocity V, which is equal to linear velocity of a tool, as follows i-
CO
(4)
co. When the processing becomes unstable due to slipping, coefficient i--->ooand the forging ceases to rotate. Consideration of moments of forces acting axially on the formed product has led A.F.Lisoczkin, as first, to determining the following formula on the maximum reduction obtained in given conditions of friction [22] 2
~r
~t
do
1+ ro R
(5)
Zb. PaterI Journal of Materials Processing Technology 60 (1996) 549-554
553
-1
stable stage
(a)
(b)
(c)
(d)
~///A
t\\',
-
""/
;////5
transient stage "///////A
Fig.7. Distribution of mean stress in cross-sections of forgings being subject to rotational compression with: p,'=0.3 and a) 8=1.05; b) 8=1.10; c) t?~l.15; d) &:l.20.
References
I%~t~x ~ /,
y,.y////z -////r// unstable stage
Fig.6. The course of rotational compression processing of lead specimen, with (~=1.16 and g=0.3.
that fields of negative stresses tr= in cross-section of forging become reduced when t$ increases, whereas they attain extreme values in the near surface zone. This is an evidence that in central zone tensile stresses tr, are predominant over compressing stresses a~. The third, normal stress try, according to the theory of plane state of strain equals to the mean stress
trm-
4. Conclusions
[ 1] W. Johnson, Engnr, 7 (1958) 348. [2] A.I. Tselikov, W. M. Lugovski and E. M. Tretiakov, Rus. Eng. d., 7 (1961) 44. [3] C.H. Lee and S. Kobyashi, d. Eng. Ind., Trans. ASME, (1971) 445. [4] L.S. Agamirzian and Dz. M. Lomsadze, lzv. VUZ, Cern. Metall. No. 11 (1969) 114. [5] Y. Saito and T. Higashino, J. dSTP, 18 (1977) 120. [6] Zb. Pater and W. Werofiski, Obr6bka Plast. No. 1 (1994) 17. [7] V. 'ira. Osadczij and I. G. Getia, Izv. VUZ, Cern. Metall. No. 5 (1970) 81. [8] M. Hayama, d. dSTP, 15 (1974) 141. [9] K. H. Na and N. S. Cho, d. Mech. Work. Technol., 19 (1989) 211. [10] K. H. Na, N. S. Cho and J. H. Kim, Proc. 4th Int. Cot~ Technology of Plasticity, Beijing, China Sept. 5-9, (1993) v.1 251. [111 M. Hayama,3. dSTP, 16 (1975) 1148. [12] G. Ya. Goon, Mathematical Modelling of Metal Forming Processes, MetaUurgia, Moscow, 1983. [13] A. I. Tselikov, Rolling Machines, Mashgiz, Moscow, 1948.
Application of commercial system FORM-2D enabled performing a theoretical analysis of rotational compression of rods. Presented in the paper, theoretical solutions concerning the width of contact area, contact pressure and conditions of stability, remain in a good agreement with experimental data published in references. A more thorough analysis of rotational compression is expected using three- dimensional code.
[14] V. S. Smimov, V. P. Anisiforov and M. V. Vasilychkov, Cross Rolling in Manufactoring, Mashgiz, Moscow, 1957. [15] Zb. Pater and W. Werofiski, J. Mater. Process. Technol. 45 (1994) 105. [16] M. Hayama, Bulletin of Fac. Engng. Yokohama Nat. Univ., 23 (1974) 83.
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[17] Y. Kasuga, Japan Soc. Mech. Engr., (197 l) 73. [18] W. A. Kluszin, E. M. Makushok and V. Ya. Shukhl, Development of Cross Wedge Rolling, Science and Technology, Minsk, 1982. [19] Zb. Pater and W. Werofiski, Theoretical Basis of Cross Wedge Rolling Processes, LTN, Lublin, 1995.
[20] Y. Kasuga, Annals of the CIRP, 22 (1974) 93. [21] G. V. Andreev (Ed.), Cross Wedge Rolling, Science and Technology, Minsk, 1974. [22] A. F. Lisockin, Stal, No. 6 (1946). [23] G. Thompsen and J. B. Hawkyard, Proc. 1st Int. Conf. on Rotary Metalworking Processes, London,U.K., (1979) 171.
