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Experimental investigation of the strain state in the ring-forging process
Journal o[ Mechanical Working Technology, 14 (1987) 309-324
309
Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
E X P E R I M E N T A L I N V E S T I G A T I O N OF T H E S T R A I N S T A T E IN THE RING-FORGING PROCESS
J. KOWALSKI, B. HODERNY
Institute of Ferrous Metallurgy, ul.K.Miarki 12/14, 44-100 Gliwice (Poland) and Z. MALINOWSKI
Department of Metallurgy, University of Mining and Metallurgy, Al. Mickiewicza 30, 30-059 Krak6w (Poland) (Received May 5, 1986; accepted October 6, 1986)
Industrial Summary A method for calculating the strain state during three-dimensional plastic deformation is presented. The method is based on the measurement of the co-ordinates of net nodes marked on the surface to be investigated: in the present case the method is applied to the ring-forging process. Specimens were forged between the mandrel and a fiat die and also between the mandrel and a die of inverted V-shape with included angle (0= 135 °. The results obtained in the tests allow an assessment of the influence of the relative width of the specimen, ~ ( ~ = original width of the ring divided by its original outer diameter), and the relative diameter of the mandrel, $ (5 =diameter of the mandrel divided by the I.D. of the ring), on the strain state in the deformation zone. A fiat die and a relative diameter of the mandrel of J i>0.8 afford less inhomogenity of strain and less irregularity of the cross-section in comparison with the results of using the inverted Vshaped die.
Notation ai
bij
Bo DT
D~o, Dwo D~i, Owl ho hki ui xi
Xi OL
factors of approximating functions strain tensor initial width of the specimen diameter of the mandrel initial outer and inner diameter of the specimen, respectively outer and inner diameter of the specimen in pass No. i, respectively initial wall thickness wall thickness after forging in pass No. i displacement of any point of a body co-ordinates of point P in the B position local co-ordinates of point P relative width of the specimen = Bo/Dzo
0378-3804/87/$03.50
© 1987 Elsevier Science Publishers B.V.
310 7 J
DS IVD
reduction factor relative diameter of the mandrel = DT/Dwo effective strain co-ordinates of point P in the B' position die angle flat die inverted V-shaped die
1. Introduction
The manufacture of rings by free-forging processes is used where the rings are of large dimensions, the development of the technology having resulted in there being many orders for this type of forging. The rings are manufactured by forging on a mandrel from pre-rings, the outer and inner diameter of which are greater at the conclusion of the forging process. In comparison with those for upsetting or drawing down, there have been only a few papers published on the ring-forging process. The first description of this process is presented in Ref. [1], where the results of the work were obtained both from experimental investigation and from observation of actual processes. A quantitative description of the strain state has been presented by Okhrimenko [ 2 ]. On the basis of observation of the actual ring-forging processes, it was noted that this process is very complex [ 3 ] and it was suggested that the incorrect selection of parameters such as the reduction of thickness, the temperature of the ring, and the size and shape of the die may lead to defects in the ring. The removal of such defects is very expensive because it requires additional heating and a greater time of forging and in some cases it may be simpler just to scrap the forging. These defects are the development of an oval shape, the presence of warping and incorrect dimensions of the forging. Due to the complicated mathematical description involved, the calculation of the strain state in the ring-forging process is very difficult: the widely used logarithmic strain (true strain) may be employed for principal directions only. Measurement of the net nodes on a small area and in several directions enables the calculation of the strain state. This method has been used by Szyndler and Szczepanik [ 4 ], where from the results of calculation these authors concluded that an inverted V-shaped die of 135 ° included angle enables the securing of a more uniform strain that would arise from the use of a flat die. Presented in this paper is the strain state, in three-dimensional co-ordinates, during the free-forging process. The ODK3 programme [ 5 ], designed for the calculation of three-dimensional problems, has been employed: the basis of calculation by the ODK3 programme is the measurement of the co-ordinates of net nodes, which allows the calculation of the magnitude of deformation. This programme determines the influence of selected technical parameters on the strain state in the process.
311
2. Method of calculation o f the strain state
It is assumed that due to the action of external forces, the body changes its position from B to B' (Fig. 1 (a) ). At the same time the shape and volume of this body change. The relationship between the co-ordinates of any point P in the B position of the body and the B' position of the body is given by the equation [ 6 ] :
~i=~i(xl,x2,x3)
i=1,2,3
(1)
It can be concluded from the continuity condition of the body that ~i functions should be of C 1 class at least, and that the determinant of transformation (1) :
(cg~i/Oxj)
D=det
(2)
QI
Xz "~"X'¢
b)
~X 2
% xt
Fig. 1. Principle of the method of calculation: (a) Scheme of deformation; (b) Local co-ordinates X, associated with point P.
312 should not be equal to zero. A displacement of any point of the body is described by the vector: ui = ~ i - x i
(3)
which gives ~i : X i ~-Ui( X1, X2, X3)
(4)
Equation (4) presents the relationship between co-ordinates xl, x2, x3 and ~1, ~2, ~s of any point of the body. The strain may be described by the tensor [ 6 ] : Oxk Oxh Knowledge of the functions ui (xl, x2, x3) in eqn. (4) is necessary for the calculation of the strain at any point P in the body B. These functions may be determined by approximation of the displacement field ui (xl, x2, x3 ) around point P by polynomials, as follows [ 7 ] : a3X~l + a4X1X2 + a~X2 + ae,X~ + aTX3 + + a8X1X3 + a9X2X3 u2 = a~o + aHX~ + a~2X~ + a~3XIX2 + a~4X2 + a~,Y~ + + a~6X3 + aITX~X3 + a~sX2X~ u3 = a~9 + a2oX1 + a2~X2 + a22X3 + a23X~X3 + a24X2X3 Ul -~ a l +a2X1 +
(5 )
where X~, X2, X3 are the local co-ordinates conjugated with point P, described by equations: Xl = x i - x °
(6)
where x ° are the co-ordinates of point P (Fig. 1 (b)). It was assumed that during plastic deformation the material is incompressible: displacement field eqn. (5) should therefore meet the following condition [ 8 ] : D = d e t (O~/c)x i) =1
(7)
The solution of eqn. ( 7 ) yields the relationship describing the value of the a22 parameter: a22 -
l +a29[aT(1+a14)--asal~] +a21 [a16(l+a2) --allaT] - 1 (1+ a2) (1+ a~t) - a~a~
(8)
Other parameters were determined using the least-square method to minimize the functional: 2
o
2
2
1
313 In this relation w ~ is the displacement of any point of body B in the surrounding point P. w ~ was obtained from the measurement of the co-ordinates of the nodes, before deformation x/~ and after deformation ~/k: w~ = x ~ - ~
i = 1,2,3;
k = 1 . . . . ,9
Logarithmic strain was calculated using the relationship between the bij tensor and the logarithmic strain tensor eii[ 8 ] : eii= 0.5 In bii
and b~ eli =0.5 arcsin ~ ~biibij
for iCj.
The effective strain, ~, was calculated from the equation: (:=
~.ij'~.ii
3. Experimental procedure Figure 2 presents the lead specimens used in the experiment. The net nodes were measured using a toolroom microscope to an accuracy of 0.01 mm, after which they were soldered together using Wood's alloy. Following deformation by the flat die (FD) and by the inverted V-shaped die (IVD), the specimens were unsoldered and the net nodes re-measured. This was repeated for reduction factors of 7 = 1.11; 1.25 and 1.42 (Figs. 3 and 4). The reduction factor is defined as 7 = ho/hki
where ho = (Dzo - D ~ o ) / 2 and hhi = ( D ~ i - D w l ) / 2 . The computer programme ODK3 for the calculation of the strain state was designed on the basis of the mathematical model presented earlier. Lines with constant value of g (~ isolines) were obtained as a result of the calculation. Effective strain ~ consists of all components of the strain tensor, and enables the estimation of the shape and the size of the deformation zone and is a measure of the deformation of the material. The value ~= 0.05 was assumed as the boundary of the plastic deformation zone. The irregularity of the enlargement and the profile of the end face were also measured after forging, at all points on the circumference of the ring.
314 THE SPECIMENS
D=o
O,.,
Bo
mm
mm
mm
130
50
32 65
THE DIES
THE MANDRELS =~
0.6
0.8
0.9
D~
Fig. 2. The specimens and the dies used in the investigation.
Fig. 3. Specimen ( o~= 0.5, 6 = 0.9 ) after deformation using the flat die (FD) with 7 = 1.42.
4. Results
4.1 Flat die It can be seen f r o m t h e d i s t r i b u t i o n of ¢ isolines (Fig. 5 ), t h a t for s p e c i m e n s with a = 0 . 2 a n d 6 = 0 . 6 d e f o r m e d with a r e d u c t i o n f a c t o r y = 1.11, i n h o m o g e n e i t y o f s t r a i n exists. T h e g r e a t e s t value of ~ (0.2) a p p e a r s close to t h e m a n d r e l whilst it falls to h a l f t h a t value o v e r t h e middle o f t h e cross-section. F o r t h e g r e a t e r r e d u c t i o n f a c t o r 7 = 1.25, t h e i n h o m o g e n e i t y o f ~- increases. E f f e c t i v e s t r a i n ~= 0.1 a p p e a r s close to t h e die a n d t h e g r e a t e s t value o f e-again arises
315
Fig. 4. Specimen ( a = 0.5, J = 0.9) after deformation using the inverted V-shaped die (IVD) with 7=1.42.
~l=0.1.--t
t-'--~='1 ~t~-~/ l'= 1.11
• /
/
" ~, = 1.11
~0.I§~
~= 1.25
~'= 1.25
.~
'I : 1,4z
•r = 1.~2
Fig. 5. Isolines of effective strain ~ for: a = 0.25; ~ = 0.6; FD. Fig. 6. Isolines of effective strain ~for: a = 0 . 2 5 ; 5 =0.8; FD.
~ i / ~
316 near to the mandrel. Increasing the redu.ction factor to 7--1.42 effects the increase of ¢ to 0.3 near to the die and to 0.6 near to the mandrel. The isoline distributions confirm t h a t the metal is very heavily deformed by the mandrel. It was also noted t h a t the distribution and the value of ¢ are virtually unchanged when the relative diameter of the mandrel increases to J = 0:8 for a reduction of 7 = 1.11 ( Fig. 6). The effective strain was ¢= 0.15 near to the die and ~= 0.3 near to the mandrel for 7 = 1.25, W h e n the reduction factor increases to 7 = 1.42, the distribution of ¢ over the cross-section changes, the effect of the die being stronger in this case, as indicated by £-- 0.3 near to the die. However, the propagation of strain caused by the die does not extend over the whole cross-section of the material, as the ~ value decreases to 0.2 at the middle of the cross-section and t h e n increases to (--0.5 near to the mandrel.
".2
0. l..__~ I
7
l "-" 1.11
d
~=1.~
i
z5
-~
.--o,~,f / --# ,~. ":£tl I ',~
"i': 1.2s
't "- 1.25
~
i
/-
= 1.4.2
,
f
"1' = 1.~,2
MJz~'z
Fig. 7. Isolinesof effectivestrain (for: ~ = 0.25; J--0.9; FD. Fig. 8. Isolinesof effectivestrain ~for: ~ = 0.5; J = 0.6; FD.
\ .)"
317
Change of the strain distribution is well seen for 5 = 0.9 ( Fig. 7 ). For a reduction factor of y = 1.11, the greatest value of e- (0.2) exists at the middle of the cross-section. A value of effective strain of ( = 0.1 appears both near to the die and near to the mandrel. The largest value of ( ( 0.3 ) exists also in the middle of the section for 7 = 1.25. The mandrel and the die equally affect the value of ~, which is about 0.15. W h e n the reduction factor increases to 1.42, the inhomogeneity of the strain decreases, the value of g changing from 0.25 near to the die and 0.3 near to the mandrel to ( = 0.35 at the middle of the section. Analysis of the distribution of isolines for specimens where ~ = 0.5 (Figs. 8-10) leads to similar conclusions as for the case where ~ = 0.25. Comparison of the isolines on the longitudinal sections for various reductions enables the observation that the greatest value of strain ( (-- 0.4) appears on the end face
t q L - o.,4 4T- O.i-.,..~
& fS----~
O.
~=1.11
~,= 1.11
i
'~"
~
i
1['=1.25
~.= 1.25
'..~" o.z~ ~ 7'~,~ |
/-o.
~ -----~,
o. J T = 1.4.2
"°~,\1((I I '¢
./"
I--:-o,-/! ~----~-~-
Z' = 1.42
Fig. 9. Isolines o f effective s t r a i n ( f o r : ~ = 0.5; J = 0.8; FD. Fig. 10. Isolines o f effective s t r a i n £ for: ~ = 0.5; J = 0.9; FD.
~z4.~ I
318
for c~= 0.25 and 6 - 0.9 with 7 = 1.42, while for a = 0.5 the isolines are parallel to the mandrel.
4.2 Inverted V-shaped die It can be seen from the analysis of + isolines for specimen dimensions = 0.25, 6 = 0.6 (Fig. 11 ) t h a t a reduction of 7 = 1.11 is associated with a large inhomogeneity of strain; the value of ~ changes from 0.2 near to the mandrel to 0.05 near to the die. W h e n the reduction factor increases to 7 = 1.25 the magnitude of the plastic deformation zone increases also, the value of effective strain changing from ~=0.4 near to the mandrel to £=0.1 near to the die; a value of £= 0.05 exists at the axis of symmetry. For a larger value of reduction factor, 7 = 1.42, the effective strain is £= 0.6 near to the mandrel and ~= 0.1-0.2 near to the die.
~:,
O.O&.~
/
T" 1.11
j.
T:1.25
T:I.25
~ I
'i" : 1.~2
Fig. 11. Isolines of effective s t r a i n £ for: ~ = 0.25; J = 0.6; IVD. Fig. 12. Isolines of effective s t r a i n £for: ~ = 0 . 2 5 ; 6 =0.8; IVD.
0.. ~
319
The distribution of isolines changes when the relative diameter of the mandrel increases to J = 0 . 8 for 7 = 1.11. The whole deformation zone is now in a plastic state. Similarly as for J--0.6, ( = 0 . 3 close to the mandrel and ( = 0.05-0.15 close to the die. For reduction factor 7 = 1.42 the distribution of strain changes and values £= 0.2 close to the mandrel and ~= 0.5 close to the die are observed. W h e n the relative diameter of the mandrel is 6 =0.9 and 7 = 1.11-1.24, the distribution of ~ is similar to t h a t for J = 0.8: a lesser inhomogeneity of strain is observed for 7 = 1.42. The influence of the relative diameter of the mandrel and the reduction factor on the strain distribution and on the value "of ~ for = 0.5 is similar to the case discussed previously (Figs. 14-16).
,(=1.11
f=1.25
•~ :
1.11
T = 1.25
T= 1/,2 T = 1.~2
Fig. 13. Isolines of effective strain (for: ~ =0.25; J=0.9; IVD. Fig. 14. Isolines of effective strain ~for: ~=0.50; 6 =0.6; IVD.
320
Other results of investigation of the strain state in the ring-forging process are presented in Ref. [ 9 ].
4.3 Overlapping of the zones of deformation Under industrial conditions, strokes are repeated several times with y held constant, during every pass. The zone of deformation which appears in the current stroke, overlaps that which arose from the previous stroke. A specimen was deformed by three subsequent strokes with the aim of investigation of the overlapping effect, for ~ = 0.5 and ci = 0.9. The parameter y was the same as for a single zone of deformation. The flat die in the ring-forging process was used. An analysis of the distribution of ¢-isolines showed that the value of ¢ at the
O.05-..-/I Z== O.'l~ I
o.t5,_..I
~'= 1.1'1
~=1.11
1'-- 1.25
,z,=l.~
T = 1:~2
Fig. 15. Isolines of effective strain ¢ for: ~ = 0.50; J = 0.8; IVD. Fig. 16. Isolines of effective strain ¢ for: ~ = 0.50; 5 = 0.9; I V D .
321 middle of the cross-section is similar to that for a single zone of deformation.As the isolines have a circumferential direction, knowledge of the strain distribution in a single zone of deformation therefore suffices (Fig. 17).
4.4 The irregularity of enlargement The profile of the end face was measured during the ring-forging process, the results of which are presented in Fig. 18 for FD and in Fig. 19 for IVD. The relative diameter of the mandrel, the relative width of the forging and the shape of the die have an influence on the value of the maximum enlargement. A mandrel with J = 0.6 is noted to result in large inhomogeneity in the deformation, whilst ring forging using an inverted V-shaped die promotes the irregularity effect. A mandrel with J >/0.8 results in lesser irregularity of the end face of the forging.
T= 1.11
•I,= 1.25
Fig. 17. Overlappingof the deformationzones.
322 = 0.25
-' = 0.5
/.82mm
2 . 9 6 rnm
y= 1.4f0
----]-
---~-.
_ ___ ~-
f.Z~m
f.72mm
II ¢,0
iI
i 4.18mm
J-i _
l_
0.89
¢5
Fig. 18. The shape of the end face measured after forging using the FD. 5. Conclusions
The results presented in this paper enable an estimation of the influence of the relative diameter of the mandrel, the relative width of the forging and the shape of die on the strain state during the ring-forging process. Forging with a flat die where J > 0.8 leads to the greatest deformation being at the centre of the cross-section and to slight irregularity of the end face, whilst forging where J < 0.8 leads to greater inhomogeneity of strain, the greatest value of the effective strain being observed close to the mandrel. A small diameter of the mandrel causes greater irregularity of the end face of the forging.
323 a=0.25
a=0.5
2.55,.,,
, _ 5. 3 t - , m
I~ I'=1"325 -----k
~
--
L68mm
5. fO m~
o,
=
~. 386
¢,'0 i
I
,
l I. 76 m m
Z
~: 1.4Z
- -- -i
Fig. 19. The shape of the end face measuredafter forgingusing the IVD. The inverted V-shaped die of included angle ~ = 135 ° affords a smaller inhomogeneity of the strain, but this die fosters the arising of various defects, such as the development of an oval shape and warping of the cross-section, which are difficult to remove. Moreover, for J < 0.8, a large irregularity of the end face of the forging is observed.
References 1 B.M. Baliasny, Kuznechno-Shtampovochnoe Proizvod., (2) (1961) 8. 2 Ya.M. Okhrimenko and V.A. Turin, Izv. Vyssh. Uchebn. Zaved. Chem. Metall., 5 (1975) 111.
324 3 H. Malinowski and J. Sirlczak, Hutnik, 46 (8-9) (1979) 360. 4 R. Szyndler and S. Szczepanik, Hutnik, 46 (1979) 404. 5 The Library of Subroutines, Unpublished Report of Institute of Metallurgy, Univ. of Mining and Metallurgy, Krak6w, 1983. 6 W. Nowacki, Teoria Spr~ysto~ci, PWN, Warszawa, 1970. 7 Z. Malinowski and S. Szczepanik, Obr6bka Plast., XXII (1983) 109. 8 P.I. Polukhin and A.V. Vorontsov, Strain and stress in metal forming, Metallurgia, Moskva, 1974 (in Russian). 9 J. Kowalski, D.Sc. dissertation, AGH, Krak6w, 1985 (unpublished).
309
Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
E X P E R I M E N T A L I N V E S T I G A T I O N OF T H E S T R A I N S T A T E IN THE RING-FORGING PROCESS
J. KOWALSKI, B. HODERNY
Institute of Ferrous Metallurgy, ul.K.Miarki 12/14, 44-100 Gliwice (Poland) and Z. MALINOWSKI
Department of Metallurgy, University of Mining and Metallurgy, Al. Mickiewicza 30, 30-059 Krak6w (Poland) (Received May 5, 1986; accepted October 6, 1986)
Industrial Summary A method for calculating the strain state during three-dimensional plastic deformation is presented. The method is based on the measurement of the co-ordinates of net nodes marked on the surface to be investigated: in the present case the method is applied to the ring-forging process. Specimens were forged between the mandrel and a fiat die and also between the mandrel and a die of inverted V-shape with included angle (0= 135 °. The results obtained in the tests allow an assessment of the influence of the relative width of the specimen, ~ ( ~ = original width of the ring divided by its original outer diameter), and the relative diameter of the mandrel, $ (5 =diameter of the mandrel divided by the I.D. of the ring), on the strain state in the deformation zone. A fiat die and a relative diameter of the mandrel of J i>0.8 afford less inhomogenity of strain and less irregularity of the cross-section in comparison with the results of using the inverted Vshaped die.
Notation ai
bij
Bo DT
D~o, Dwo D~i, Owl ho hki ui xi
Xi OL
factors of approximating functions strain tensor initial width of the specimen diameter of the mandrel initial outer and inner diameter of the specimen, respectively outer and inner diameter of the specimen in pass No. i, respectively initial wall thickness wall thickness after forging in pass No. i displacement of any point of a body co-ordinates of point P in the B position local co-ordinates of point P relative width of the specimen = Bo/Dzo
0378-3804/87/$03.50
© 1987 Elsevier Science Publishers B.V.
310 7 J
DS IVD
reduction factor relative diameter of the mandrel = DT/Dwo effective strain co-ordinates of point P in the B' position die angle flat die inverted V-shaped die
1. Introduction
The manufacture of rings by free-forging processes is used where the rings are of large dimensions, the development of the technology having resulted in there being many orders for this type of forging. The rings are manufactured by forging on a mandrel from pre-rings, the outer and inner diameter of which are greater at the conclusion of the forging process. In comparison with those for upsetting or drawing down, there have been only a few papers published on the ring-forging process. The first description of this process is presented in Ref. [1], where the results of the work were obtained both from experimental investigation and from observation of actual processes. A quantitative description of the strain state has been presented by Okhrimenko [ 2 ]. On the basis of observation of the actual ring-forging processes, it was noted that this process is very complex [ 3 ] and it was suggested that the incorrect selection of parameters such as the reduction of thickness, the temperature of the ring, and the size and shape of the die may lead to defects in the ring. The removal of such defects is very expensive because it requires additional heating and a greater time of forging and in some cases it may be simpler just to scrap the forging. These defects are the development of an oval shape, the presence of warping and incorrect dimensions of the forging. Due to the complicated mathematical description involved, the calculation of the strain state in the ring-forging process is very difficult: the widely used logarithmic strain (true strain) may be employed for principal directions only. Measurement of the net nodes on a small area and in several directions enables the calculation of the strain state. This method has been used by Szyndler and Szczepanik [ 4 ], where from the results of calculation these authors concluded that an inverted V-shaped die of 135 ° included angle enables the securing of a more uniform strain that would arise from the use of a flat die. Presented in this paper is the strain state, in three-dimensional co-ordinates, during the free-forging process. The ODK3 programme [ 5 ], designed for the calculation of three-dimensional problems, has been employed: the basis of calculation by the ODK3 programme is the measurement of the co-ordinates of net nodes, which allows the calculation of the magnitude of deformation. This programme determines the influence of selected technical parameters on the strain state in the process.
311
2. Method of calculation o f the strain state
It is assumed that due to the action of external forces, the body changes its position from B to B' (Fig. 1 (a) ). At the same time the shape and volume of this body change. The relationship between the co-ordinates of any point P in the B position of the body and the B' position of the body is given by the equation [ 6 ] :
~i=~i(xl,x2,x3)
i=1,2,3
(1)
It can be concluded from the continuity condition of the body that ~i functions should be of C 1 class at least, and that the determinant of transformation (1) :
(cg~i/Oxj)
D=det
(2)
QI
Xz "~"X'¢
b)
~X 2
% xt
Fig. 1. Principle of the method of calculation: (a) Scheme of deformation; (b) Local co-ordinates X, associated with point P.
312 should not be equal to zero. A displacement of any point of the body is described by the vector: ui = ~ i - x i
(3)
which gives ~i : X i ~-Ui( X1, X2, X3)
(4)
Equation (4) presents the relationship between co-ordinates xl, x2, x3 and ~1, ~2, ~s of any point of the body. The strain may be described by the tensor [ 6 ] : Oxk Oxh Knowledge of the functions ui (xl, x2, x3) in eqn. (4) is necessary for the calculation of the strain at any point P in the body B. These functions may be determined by approximation of the displacement field ui (xl, x2, x3 ) around point P by polynomials, as follows [ 7 ] : a3X~l + a4X1X2 + a~X2 + ae,X~ + aTX3 + + a8X1X3 + a9X2X3 u2 = a~o + aHX~ + a~2X~ + a~3XIX2 + a~4X2 + a~,Y~ + + a~6X3 + aITX~X3 + a~sX2X~ u3 = a~9 + a2oX1 + a2~X2 + a22X3 + a23X~X3 + a24X2X3 Ul -~ a l +a2X1 +
(5 )
where X~, X2, X3 are the local co-ordinates conjugated with point P, described by equations: Xl = x i - x °
(6)
where x ° are the co-ordinates of point P (Fig. 1 (b)). It was assumed that during plastic deformation the material is incompressible: displacement field eqn. (5) should therefore meet the following condition [ 8 ] : D = d e t (O~/c)x i) =1
(7)
The solution of eqn. ( 7 ) yields the relationship describing the value of the a22 parameter: a22 -
l +a29[aT(1+a14)--asal~] +a21 [a16(l+a2) --allaT] - 1 (1+ a2) (1+ a~t) - a~a~
(8)
Other parameters were determined using the least-square method to minimize the functional: 2
o
2
2
1
313 In this relation w ~ is the displacement of any point of body B in the surrounding point P. w ~ was obtained from the measurement of the co-ordinates of the nodes, before deformation x/~ and after deformation ~/k: w~ = x ~ - ~
i = 1,2,3;
k = 1 . . . . ,9
Logarithmic strain was calculated using the relationship between the bij tensor and the logarithmic strain tensor eii[ 8 ] : eii= 0.5 In bii
and b~ eli =0.5 arcsin ~ ~biibij
for iCj.
The effective strain, ~, was calculated from the equation: (:=
~.ij'~.ii
3. Experimental procedure Figure 2 presents the lead specimens used in the experiment. The net nodes were measured using a toolroom microscope to an accuracy of 0.01 mm, after which they were soldered together using Wood's alloy. Following deformation by the flat die (FD) and by the inverted V-shaped die (IVD), the specimens were unsoldered and the net nodes re-measured. This was repeated for reduction factors of 7 = 1.11; 1.25 and 1.42 (Figs. 3 and 4). The reduction factor is defined as 7 = ho/hki
where ho = (Dzo - D ~ o ) / 2 and hhi = ( D ~ i - D w l ) / 2 . The computer programme ODK3 for the calculation of the strain state was designed on the basis of the mathematical model presented earlier. Lines with constant value of g (~ isolines) were obtained as a result of the calculation. Effective strain ~ consists of all components of the strain tensor, and enables the estimation of the shape and the size of the deformation zone and is a measure of the deformation of the material. The value ~= 0.05 was assumed as the boundary of the plastic deformation zone. The irregularity of the enlargement and the profile of the end face were also measured after forging, at all points on the circumference of the ring.
314 THE SPECIMENS
D=o
O,.,
Bo
mm
mm
mm
130
50
32 65
THE DIES
THE MANDRELS =~
0.6
0.8
0.9
D~
Fig. 2. The specimens and the dies used in the investigation.
Fig. 3. Specimen ( o~= 0.5, 6 = 0.9 ) after deformation using the flat die (FD) with 7 = 1.42.
4. Results
4.1 Flat die It can be seen f r o m t h e d i s t r i b u t i o n of ¢ isolines (Fig. 5 ), t h a t for s p e c i m e n s with a = 0 . 2 a n d 6 = 0 . 6 d e f o r m e d with a r e d u c t i o n f a c t o r y = 1.11, i n h o m o g e n e i t y o f s t r a i n exists. T h e g r e a t e s t value of ~ (0.2) a p p e a r s close to t h e m a n d r e l whilst it falls to h a l f t h a t value o v e r t h e middle o f t h e cross-section. F o r t h e g r e a t e r r e d u c t i o n f a c t o r 7 = 1.25, t h e i n h o m o g e n e i t y o f ~- increases. E f f e c t i v e s t r a i n ~= 0.1 a p p e a r s close to t h e die a n d t h e g r e a t e s t value o f e-again arises
315
Fig. 4. Specimen ( a = 0.5, J = 0.9) after deformation using the inverted V-shaped die (IVD) with 7=1.42.
~l=0.1.--t
t-'--~='1 ~t~-~/ l'= 1.11
• /
/
" ~, = 1.11
~0.I§~
~= 1.25
~'= 1.25
.~
'I : 1,4z
•r = 1.~2
Fig. 5. Isolines of effective strain ~ for: a = 0.25; ~ = 0.6; FD. Fig. 6. Isolines of effective strain ~for: a = 0 . 2 5 ; 5 =0.8; FD.
~ i / ~
316 near to the mandrel. Increasing the redu.ction factor to 7--1.42 effects the increase of ¢ to 0.3 near to the die and to 0.6 near to the mandrel. The isoline distributions confirm t h a t the metal is very heavily deformed by the mandrel. It was also noted t h a t the distribution and the value of ¢ are virtually unchanged when the relative diameter of the mandrel increases to J = 0:8 for a reduction of 7 = 1.11 ( Fig. 6). The effective strain was ¢= 0.15 near to the die and ~= 0.3 near to the mandrel for 7 = 1.25, W h e n the reduction factor increases to 7 = 1.42, the distribution of ¢ over the cross-section changes, the effect of the die being stronger in this case, as indicated by £-- 0.3 near to the die. However, the propagation of strain caused by the die does not extend over the whole cross-section of the material, as the ~ value decreases to 0.2 at the middle of the cross-section and t h e n increases to (--0.5 near to the mandrel.
".2
0. l..__~ I
7
l "-" 1.11
d
~=1.~
i
z5
-~
.--o,~,f / --# ,~. ":£tl I ',~
"i': 1.2s
't "- 1.25
~
i
/-
= 1.4.2
,
f
"1' = 1.~,2
MJz~'z
Fig. 7. Isolinesof effectivestrain (for: ~ = 0.25; J--0.9; FD. Fig. 8. Isolinesof effectivestrain ~for: ~ = 0.5; J = 0.6; FD.
\ .)"
317
Change of the strain distribution is well seen for 5 = 0.9 ( Fig. 7 ). For a reduction factor of y = 1.11, the greatest value of e- (0.2) exists at the middle of the cross-section. A value of effective strain of ( = 0.1 appears both near to the die and near to the mandrel. The largest value of ( ( 0.3 ) exists also in the middle of the section for 7 = 1.25. The mandrel and the die equally affect the value of ~, which is about 0.15. W h e n the reduction factor increases to 1.42, the inhomogeneity of the strain decreases, the value of g changing from 0.25 near to the die and 0.3 near to the mandrel to ( = 0.35 at the middle of the section. Analysis of the distribution of isolines for specimens where ~ = 0.5 (Figs. 8-10) leads to similar conclusions as for the case where ~ = 0.25. Comparison of the isolines on the longitudinal sections for various reductions enables the observation that the greatest value of strain ( (-- 0.4) appears on the end face
t q L - o.,4 4T- O.i-.,..~
& fS----~
O.
~=1.11
~,= 1.11
i
'~"
~
i
1['=1.25
~.= 1.25
'..~" o.z~ ~ 7'~,~ |
/-o.
~ -----~,
o. J T = 1.4.2
"°~,\1((I I '¢
./"
I--:-o,-/! ~----~-~-
Z' = 1.42
Fig. 9. Isolines o f effective s t r a i n ( f o r : ~ = 0.5; J = 0.8; FD. Fig. 10. Isolines o f effective s t r a i n £ for: ~ = 0.5; J = 0.9; FD.
~z4.~ I
318
for c~= 0.25 and 6 - 0.9 with 7 = 1.42, while for a = 0.5 the isolines are parallel to the mandrel.
4.2 Inverted V-shaped die It can be seen from the analysis of + isolines for specimen dimensions = 0.25, 6 = 0.6 (Fig. 11 ) t h a t a reduction of 7 = 1.11 is associated with a large inhomogeneity of strain; the value of ~ changes from 0.2 near to the mandrel to 0.05 near to the die. W h e n the reduction factor increases to 7 = 1.25 the magnitude of the plastic deformation zone increases also, the value of effective strain changing from ~=0.4 near to the mandrel to £=0.1 near to the die; a value of £= 0.05 exists at the axis of symmetry. For a larger value of reduction factor, 7 = 1.42, the effective strain is £= 0.6 near to the mandrel and ~= 0.1-0.2 near to the die.
~:,
O.O&.~
/
T" 1.11
j.
T:1.25
T:I.25
~ I
'i" : 1.~2
Fig. 11. Isolines of effective s t r a i n £ for: ~ = 0.25; J = 0.6; IVD. Fig. 12. Isolines of effective s t r a i n £for: ~ = 0 . 2 5 ; 6 =0.8; IVD.
0.. ~
319
The distribution of isolines changes when the relative diameter of the mandrel increases to J = 0 . 8 for 7 = 1.11. The whole deformation zone is now in a plastic state. Similarly as for J--0.6, ( = 0 . 3 close to the mandrel and ( = 0.05-0.15 close to the die. For reduction factor 7 = 1.42 the distribution of strain changes and values £= 0.2 close to the mandrel and ~= 0.5 close to the die are observed. W h e n the relative diameter of the mandrel is 6 =0.9 and 7 = 1.11-1.24, the distribution of ~ is similar to t h a t for J = 0.8: a lesser inhomogeneity of strain is observed for 7 = 1.42. The influence of the relative diameter of the mandrel and the reduction factor on the strain distribution and on the value "of ~ for = 0.5 is similar to the case discussed previously (Figs. 14-16).
,(=1.11
f=1.25
•~ :
1.11
T = 1.25
T= 1/,2 T = 1.~2
Fig. 13. Isolines of effective strain (for: ~ =0.25; J=0.9; IVD. Fig. 14. Isolines of effective strain ~for: ~=0.50; 6 =0.6; IVD.
320
Other results of investigation of the strain state in the ring-forging process are presented in Ref. [ 9 ].
4.3 Overlapping of the zones of deformation Under industrial conditions, strokes are repeated several times with y held constant, during every pass. The zone of deformation which appears in the current stroke, overlaps that which arose from the previous stroke. A specimen was deformed by three subsequent strokes with the aim of investigation of the overlapping effect, for ~ = 0.5 and ci = 0.9. The parameter y was the same as for a single zone of deformation. The flat die in the ring-forging process was used. An analysis of the distribution of ¢-isolines showed that the value of ¢ at the
O.05-..-/I Z== O.'l~ I
o.t5,_..I
~'= 1.1'1
~=1.11
1'-- 1.25
,z,=l.~
T = 1:~2
Fig. 15. Isolines of effective strain ¢ for: ~ = 0.50; J = 0.8; IVD. Fig. 16. Isolines of effective strain ¢ for: ~ = 0.50; 5 = 0.9; I V D .
321 middle of the cross-section is similar to that for a single zone of deformation.As the isolines have a circumferential direction, knowledge of the strain distribution in a single zone of deformation therefore suffices (Fig. 17).
4.4 The irregularity of enlargement The profile of the end face was measured during the ring-forging process, the results of which are presented in Fig. 18 for FD and in Fig. 19 for IVD. The relative diameter of the mandrel, the relative width of the forging and the shape of the die have an influence on the value of the maximum enlargement. A mandrel with J = 0.6 is noted to result in large inhomogeneity in the deformation, whilst ring forging using an inverted V-shaped die promotes the irregularity effect. A mandrel with J >/0.8 results in lesser irregularity of the end face of the forging.
T= 1.11
•I,= 1.25
Fig. 17. Overlappingof the deformationzones.
322 = 0.25
-' = 0.5
/.82mm
2 . 9 6 rnm
y= 1.4f0
----]-
---~-.
_ ___ ~-
f.Z~m
f.72mm
II ¢,0
iI
i 4.18mm
J-i _
l_
0.89
¢5
Fig. 18. The shape of the end face measured after forging using the FD. 5. Conclusions
The results presented in this paper enable an estimation of the influence of the relative diameter of the mandrel, the relative width of the forging and the shape of die on the strain state during the ring-forging process. Forging with a flat die where J > 0.8 leads to the greatest deformation being at the centre of the cross-section and to slight irregularity of the end face, whilst forging where J < 0.8 leads to greater inhomogeneity of strain, the greatest value of the effective strain being observed close to the mandrel. A small diameter of the mandrel causes greater irregularity of the end face of the forging.
323 a=0.25
a=0.5
2.55,.,,
, _ 5. 3 t - , m
I~ I'=1"325 -----k
~
--
L68mm
5. fO m~
o,
=
~. 386
¢,'0 i
I
,
l I. 76 m m
Z
~: 1.4Z
- -- -i
Fig. 19. The shape of the end face measuredafter forgingusing the IVD. The inverted V-shaped die of included angle ~ = 135 ° affords a smaller inhomogeneity of the strain, but this die fosters the arising of various defects, such as the development of an oval shape and warping of the cross-section, which are difficult to remove. Moreover, for J < 0.8, a large irregularity of the end face of the forging is observed.
References 1 B.M. Baliasny, Kuznechno-Shtampovochnoe Proizvod., (2) (1961) 8. 2 Ya.M. Okhrimenko and V.A. Turin, Izv. Vyssh. Uchebn. Zaved. Chem. Metall., 5 (1975) 111.
324 3 H. Malinowski and J. Sirlczak, Hutnik, 46 (8-9) (1979) 360. 4 R. Szyndler and S. Szczepanik, Hutnik, 46 (1979) 404. 5 The Library of Subroutines, Unpublished Report of Institute of Metallurgy, Univ. of Mining and Metallurgy, Krak6w, 1983. 6 W. Nowacki, Teoria Spr~ysto~ci, PWN, Warszawa, 1970. 7 Z. Malinowski and S. Szczepanik, Obr6bka Plast., XXII (1983) 109. 8 P.I. Polukhin and A.V. Vorontsov, Strain and stress in metal forming, Metallurgia, Moskva, 1974 (in Russian). 9 J. Kowalski, D.Sc. dissertation, AGH, Krak6w, 1985 (unpublished).