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Prediction of deformation and homogeneity in rim-disc forging
Journal of Mechanical Working Technology, 4 (1980) 145--154 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
145
PREDICTION OF DEFORMATION AND HOMOGENEITY IN R I M - D I S C FORGING
P. HARTLEY, C.E.N. STURGESS and G.W. ROWE Department of Mechanical Engineering, University of Birmingham, P.O. Box 363, Birmingham B 15 2TT (England) (Received November 12, 1979; accepted in revised form February 1, 1980)
Industrial S u m m a r y It is often important in practical forging and other mechanical working processes to be able to predict the detailed mechanical and metallurgical properties of the product. Hitherto this has always involved prior experience or expensive pre-production trials. Theoretical analyses have always made assumptions about material properties that are valid for force predictions but too crude for detailed quality assessment. The relatively new finite-element technique allows the local properties of the billet to be included in predictions of internal distortion and overall hardening. These can, in principle, be used as a guide to production routes for a given final shape, thus greatly reducing lead-time, and also serve as indicators of likely service performance. This paper describes a first attempt to predict such important features as internal hardness distribution and excessive local grain growth. An axisymmetric rim-disc forging is chosen and the results are compared with a cold-forged aluminium product and an industrially produced aluminium alloy extrusion, showing close correlation.
Introduction I n h o m o g e n e o u s d e f o r m a t i o n can o c c u r in all forgings, and leads t o r e d u c e d fatigue life or p r e m a t u r e failure o f the p r o d u c t in service. A prior k n o w l e d g e o f the p r o b a b l e d e f o r m a t i o n characteristics in a particular process m a y be used as a guide in o r d e r t o avoid d e t r i m e n t a l features, and p r o d u c e either a h o m o g e n e o u s s t r u c t u r e or a prescribed d i s t r i b u t i o n o f properties. Various finite-element t e c h n i q u e s have been used t o s t u d y : m e t a l - f l o w [ 1 - - 3 ] , forces [ 4 ] , strain [5] and t e m p e r a t u r e [6] distributions in metalf o r m i n g problems, and have been reviewed elsewhere [ 7, 8]. T h e p r e s e n t p a p e r a d o p t s an i n c r e m e n t a l elastic-plastic a p p r o a c h [ 9 , 1 0 ] in w h i c h the influence o f interface friction is i n c l u d e d b y the a d d i t i o n o f a n a r r o w layer o f elements on a n y surface o f t h e m o d e l w h e r e friction is p r e s e n t (see a p p e n d i x ) . The stiffnesses o f these layers are m o d i f i e d b y a f u n c t i o n o f the interface shear ratio, m. This t e c h n i q u e was s h o w n t o be valid f o r t h e analysis of the c o m p r e s s i o n o f rings in the ring-test [ 10], b u t until n o w has n o t been used f o r a n y o t h e r g e o m e t r y .
146 PUNCH
CO,,~AINEP
e
CONTAINER8
PJNCH
~:
Im~
/ ~ / / ~ / / ~:'BILLET
Fig. 1. Initial position (a) and deformed state (b) in axisymmetric rim-disc forging. Fig. 2. Finite-element mesh, including extra layer of 'friction' elements, representing one half of a meridian plane.
The rim-disc forging configuration is shown in Fig. 1 and the finite-element representation illustrated in Fig. 2. Stress-strain curve for aluminium Commercially-pure aluminium billets were machined to various height/ diameter ratios in their as-received condition. A Cook-and-Larke procedure [ 11] was used to evaluate the stress-strain curve illustrated in Fig. 3. An empirical relationship that was found to fit the experimental curve reasonably well is: = oy + 21.62 ln(~/0.00624)
(1)
where ~ and ~ are the current generalised stress and generalised strain, respectively, and ay is the initial yield stress. 200 STRESS 100 : N/mm 2
0
1
2 3 4 5 NATURAL STRAIN
6
7
8
Fig. 3. Experimental stress-strain results (A~) and representation ( - - ) used as input to finite-element program.
147
Experimental tim-disc forging The aluminium specimens, 39 m m dia. X 13.3 mm high, were p r o d u c e d in two halves with a rectangular grid etched o n t o the meridian plane of one half [ 12]. The two halves of the specimen were then relocated in the closely fitting container. Lanolin was applied to all external surfaces of the billet to provide good lubrication. The specimens were then d e f o r m e d to the required level at a very low strain-rate. After removal of the specimens from the container, the d e f o r m a t i o n pattern could be determined from observation of the d e f o r m e d grid (Fig. 4).
Fig. 4. E x p e r i m e n t a l d e f o r m a t i o n of initially square grid; (a) 10%, (b) 19%, (c) 26%: The grid lines shown are selected from an original experimental grid consisting of 38 vertical and 14 horizontal lines.
The interface shear factor m, for the lanolin lubrication used in these experiments, was determined from a ring-test using a ring-material of commercially-pure aluminium with lanolin as the lubricant [ 1]; the m-factor was f o u n d to be 0.04. The experimental hardness distributions expressed in terms of the Vickers Pyramid N u m b e r were obtained with a 5 kg load on a manual macro-hardness testing machine. Subsequent to the post-process heat t r e a t m e n t (annealing at 450°C for 2 hours), the specimens were repolished on a 600 grit disc and after cleaning were immersed in a solution of cupric chloride (125 gm) and water (1000 gm). T h e y were then washed in water and immersed in a 50% nitric acid solution to remove any of the c oppe r deposit and reveal the grain structure of the forging. The specimens were finally cleaned with acetone and dried.
148
Discussion of results Figure 5 illustrates the very good correlation found between the experimental and theoretical forces. In each case a steadily increasing force is found although the theory underestimates the actual force at the 4--8% deformation (reduction in height) range. This may be due to the theoretical assumption that the lubricant is always present, represented by the initially specified interface shear factor m. It is extremely difficult to ascertain whether full lubrication was maintained in the experiments, although with this type of forging it is likely that it was not. 2OO
I
r
[
FORCE
100
I,',N
0
i 0
5
10 15 OEFORMAT[ON
I 20
25
30
%
Fig. 5. Forging force predicted by finite-element m e t h o d ( - - ) mined ( ........>).
and e x p e r i m e n t a l l y deter-
The theoretical predictions of internal metal-flow illustrated in Fig. 6 indicate the formation of a dead metal region beneath the punch. This type of deformation pattern is frequently observed in other forging configurations
'iiiiiiiiilli'~ trait
t tl
I 1.24-~---~t
J [ I I I I I I.I~-t--~['T~.L.IMI I I I I I I I ~ ~ _ L I [ l l l l l
I
. . . . . . . .
illjiiiiill
, , , , , , , , ,
...... Jlilli' ]l[Jll,,,,
I
Fig. 6. Finite-element predictions of internal deformation; (a) 10%, (b) 16%, (c) 21%, (d) 26%.
149
where a flat punch is used. Another region subject to little deformation is at the base, near to the right-hand corner of the billet. The majority of the deformation is therefore restricted to a diagonal region running from the base-centre to the punch corner. This t y p e of deformation is clearly shown on the experimental grids in Fig. 4. The only major discrepancy between theory and experiment is in the metal-flow around the sharp punch comer. The predictions clearly underestimate the high shearing occurring just below the punch c o m e r and consequently predict a much more curved free surface than is actually found. It is thought that this restriction is partly a result of using the triangular elements in the arrangement shown in Fig. 2. A change to higher-order elements may reduce this problem.
FIG 7a
F
G
FIG 8a
O
C
F
I
C
D
FIG 7b
E FIG 8b
E
A
0
E
A
B 0
Fig. 7. Finite-element-predicted generalised strain distributions (a) 10%, (b) 26%. Strain ranges, (A) 0.61--1.03; (B) 0.46--0.60; (C) 0.31--0.45; (D) 0.16--0.30; (E) 0.11--0.15; (F) 0.06--0.10; (G) 0.04--0.05. Fig. 8. Finite-element-predicted generalised stress distributions (a) 10%, (b) 26%. Stress ranges, (A) 156--165 N/mm2; (B) 146--155; (C) 136--145; (D) 126--135; (E) 116--125; (F) 101--115; (G) 94--100.
150
This pattern of internal deformation is obviously reflected in the strain and stress distributions, shown in Figs. 7 and 8, respectively. The highest strains appear around the punch corner as it penetrates the billet and the lowest strains in the region beneath the punch and close to the lower righthand corner. The most significant results are the hardness distributions, shown in Figs. 9 and 10. These hardness values are of particular importance in assessing the
~ 30(32) 3003)
30(31) 29(32) 30(29) 40(3~ 39(36)
0(30)
37(33) 33(34) 34(34)
40(39) 35(36)
36(34) 3£(39) 36(37) 35(34) 3404)
30(30) 3002)
31(34) 32(32)
31(33) 3203)
33(35) 36(36) 37(35)
35(35) 36(35) 34(35) 33(35)
33(37) 3404)
35(34) 36(34) 32(34)
33(32) 32(32) 32(33) 32(36)
3101)
31(32)
37(40) 36(36) 35(37) 33(32) 31(32.3 29(30) 28(30) 3001)
30(33
36(39
37(37) 36(36) 34(35)
33(34)
31(31) 30(31)
Fig. 9. Comparison of finite-element-predicted hardness and experimental hardness (in parentheses) distributions (VPN) at 10% deformation.
I 37(30) 36(28) 36(29)
36(31)
40(43)
36(32) 49(45) 45(44) 40(39) 46(41) 44(43)
40(34)
41(45)
37(31) 37(30) 37(32) 39(36) 44(33) 44(41) 42(41) 40(40) 40(37)
38(35) 4O04)
4O04)
43(40
44(37) 40(37) 40(39)39(39)
39(39)
42(39) 42(38) 4307)
42(38) 40(39) 38(39) 37(34) 38(38) 38(41)
44(40)
40(39) 38(38) 3604)
44 (41) 42 (40)
3504)
36(35) 38(40)
Fig. 10. Comparison of finite-element-predicted hardness and experimental (in parentheses) distributions (VPN) at 26% deformation.
151 suitability of any c o m p o n e n t for a particular application. The theoretical hardness values were evaluated from the well known relationship [ 13], VPN = 2.967 ~
kgf/mm 2
(2)
At a reduction of height of 10% (Fig. 9), the correlation between the finiteelement prediction and the experimental distribution of the hardness values is very good, with an average variation of 3.8%. When the height reduction is increased to 26% the correlation -- although good -- is not as close, with an average variation of 8.6%. The predicted values generally appear to be higher than the experimental, especially in the highly deformed region. However, the experimental hardness values show surprisingly little change as the height reduction is increased from 10% to 26%. This may be due in part to a reduction in the strain-hardening of the aluminium due to temperature increases in the highly worked region, which the current finite-element program does not allow for. The restriction to flow around the punch corner is also likely to contribute to the high strains in the theoretical model. Despite this minor discrepancy, the actual deformation patterns and hardness distributions correlate very closely in both cases. The grain structure on the meridian plane of the specimen illustrated in Fig. 11 clearly shows the result of inhomogeneous deformation, indicated both by the experimentally deformed grids and the finite-element predictions. The corresponding predicted strain distribution is shown in Fig. 7 (b). The critical strain for large grain growth in aluminium is between 5 and 10%, so that the areas of lowest strain should therefore contain the largest grains and the band of intense shear the smallest. This arrangement is clearly revealed on the etched specimen in Fig. 11. It is very interesting to observe a closely similar pattern in a recrystallised aiuminium alloy extrusion produced under industrial conditions (Fig. 12).
Fig. 11. Etched specimen showing grain structure and approximate limits of regions with little deformation.
152
Fig. 12. Industrially produced aluminium extrusion showing regions of little deformation at the extrusion base and in the extreme sections of the annulus (courtesy of High Duty Alloys Forgings Ltd.). Conclusions T h e forging o f a disc with an annular c i r c u m f e r e n t i a l rim has b e e n examined in detail. An elastic-plastic f i n i t e - e l e m e n t m e t h o d has been used t o p r e d i c t the p u n c h forces, metal-flow, strain, stress and hardness distributions and to indicate t h e zones o f large grain growth. Close c o r r e l a t i o n was f o u n d with the e x p e r i m e n t s using c o m m e r c i a l l y p u r e a l u m i n i u m as a m o d e l material. T h e i n t r o d u c t i o n o f surface e l e m e n t layers with a stiffness m o d i f i c a t i o n f a c t o r based o n the interface shear ratio appears t o be quite successful. Rim-disc forging is typical o f processes displaying i n h o m o g e n e o u s deformation, and shows a band o f intense shear f r o m the base-centre to the p u n c h
153 c o m e r . T h e finite e l e m e n t p r e d i c t i o n s clearly indicate this t y p e o f i n h o m o geneous structure and predict with good accuracy the distribution of the p r o p e r t i e s o f the forging. A c c u r a t e d e t e r m i n a t i o n o f t h e strain a n d stress d i s t r i b u t i o n t h r o u g h o u t t h e forging c o u l d lead t o f u r t h e r d e v e l o p m e n t s o f this t y p e o f n u m e r i c a l analysis. Particularly o f i n t e r e s t is t h e p o t e n t i a l o f this a p p r o a c h f o r predicting f l o w and p o s s i b l y f r a c t u r e in m e t a l f o r m i n g processes, especially in cases w h e r e a s e c o n d o p e r a t i o n o f a d i f f e r e n t n a t u r e is p e r f o r m e d o n t h e first-stage forging.
Acknowledgements G r a t i t u d e is e x p r e s s e d t o t h e Science R e s e a r c h Council f o r f i n a n c i n g the r e s e a r c h a n d t o Mr A. D o w n i n g f o r assistance with t h e e x p e r i m e n t a l w o r k . T h a n k s are also d u e to P r o f e s s o r J.M. A l e x a n d e r f o r his v a l u a b l e c o m m e n t s o n t h e h a r d n e s s p r e d i c t i o n p r o c e d u r e , a n d t o High D u t y A l l o y s Ltd. f o r p e r m i s s i o n to r e p r o d u c e Fig. 12.
References 1 P. Hartley, C.E.N. Sturgess and G.W. Rowe, Proc. 20th Int. MTDR Conf., Birmingham (1979), p. 157, Macmillan, London (1980). 2 E.H. Lee, R.L. Mallett and W.H. Yang, Comp. Meth. App. Mech. Eng., 10 (1977) 339. 3 H. Matsumoto, S.I. Oh and S. Kobayashi, Proc. 18th Int. MTDR Conf., Imperial College, London (1977), p. 3, Macmillan, London (1978). 4 O.C. Zienkiewicz, P.C. Jain and E. Onate, Int. J. Solids Struct., 14 (1978) 15. 5 S.N. Shah and S. Kobayashi, Proc. 15th MTDR Conf., Birmingham (1974), p. 603, Macmillan, London (1975). 6 F.W. Sharman, Electricity Council Research Report R581 (1975). 7 J.M. Alexander and J.W.H. Price, Proc. 18th MTDR Conf., London (1977), p. 267, Macmillan, London {1978). 8 H. Takahashi and S. Kobayashi, Proc. 5th NAMRC Conf., Amherst, Massachusetts (1977), p. 87. 9 P. Hartley, Ph.D. Thesis, The University of Birmingham, U.K. (1979). 10 P. Hartley, C.E.N. Sturgess and G.W. Rowe, Int. J. Mech. Sci., 21 (1979) 301. 11 M. Cook and E.C. Larke, J. Inst. Met., 71 (1945) 371. 12 G.W. Rowe, I.M. Desai and H.S. Shin, Proc. 15th Int. MTDR Conf., Birmingham (1974), p. 417, Macmillan, London (1975). 13 F.P. Bowden and D. Tabor, The Friction and Lubrication of Solids, Oxford University Press, 1964. Appendix T h e f i n i t e - e l e m e n t m e s h illustrated in Fig. 1 o f the t e x t c o n s i s t e d o f 8 8 8 a x i - s y m m e t r i c t r i a n g u l a r e l e m e n t s w i t h 506 n o d a l points. This p r o g r a m req u i r e d 2 . 7 5 s e c o n d s o f c o m p u t e r t i m e o n a CDC 7 6 0 0 c o m p u t e r p e r 0.25% d e f o r m a t i o n i n c r e m e n t . An i n c r e m e n t a l t y p e o f elastic-plastic analysis was
154 adopted in which no iterations were involved. In this approach the incremental strain is related to the incremental nodal point displacements in the form {Ae} = [B]{AS}
(A1)
where {Ae} is the strain vector, {AS} is the displacement vector and [B] is the co-ordinate matrix formulated by assuming a linear-velocity displacement of the nodal points. The incremental stress is related to incremental strain in the form, (Ao} = [DI{Ae}
(A2)
where (Ao} is the stress vector and [D] is the elastic-plastic stress-strain matrix formulated from the Prandtl-Reuss equations for deviatoric strain and stress increments. Applying a virtual-work formulation will yield, {&Q}n = [g]n_l{AS}n
(A3)
where {AQ} is the vector of applied forces, n is the increment number and [K] is the stiffness matrix evaluated from, [g] = [B]T[D] [B]v
(A4)
where v is the element volume. The resulting large number of simultaneous equations are solved by a Gauss direct-elimination technique. The influence of friction is included in the model by the addition of a narrow layer of elements on any interface where friction is present. The stiffnesses of these layers were then modified by a function, 9, of the interface shear ratio, m, i.e. {3 = r n / ( 1 - m ) .
(A5)
This required the re-formulation of eqn. (A3) to, {AQ} n = ([K]Bn_ 1 +~[g]Ln_l)(AS}n
(A6)
where B and L indicate the billet and layer, respectively. More complete details of the finite-element and friction techniques used here can be found in Refs. 1 and 10.
145
PREDICTION OF DEFORMATION AND HOMOGENEITY IN R I M - D I S C FORGING
P. HARTLEY, C.E.N. STURGESS and G.W. ROWE Department of Mechanical Engineering, University of Birmingham, P.O. Box 363, Birmingham B 15 2TT (England) (Received November 12, 1979; accepted in revised form February 1, 1980)
Industrial S u m m a r y It is often important in practical forging and other mechanical working processes to be able to predict the detailed mechanical and metallurgical properties of the product. Hitherto this has always involved prior experience or expensive pre-production trials. Theoretical analyses have always made assumptions about material properties that are valid for force predictions but too crude for detailed quality assessment. The relatively new finite-element technique allows the local properties of the billet to be included in predictions of internal distortion and overall hardening. These can, in principle, be used as a guide to production routes for a given final shape, thus greatly reducing lead-time, and also serve as indicators of likely service performance. This paper describes a first attempt to predict such important features as internal hardness distribution and excessive local grain growth. An axisymmetric rim-disc forging is chosen and the results are compared with a cold-forged aluminium product and an industrially produced aluminium alloy extrusion, showing close correlation.
Introduction I n h o m o g e n e o u s d e f o r m a t i o n can o c c u r in all forgings, and leads t o r e d u c e d fatigue life or p r e m a t u r e failure o f the p r o d u c t in service. A prior k n o w l e d g e o f the p r o b a b l e d e f o r m a t i o n characteristics in a particular process m a y be used as a guide in o r d e r t o avoid d e t r i m e n t a l features, and p r o d u c e either a h o m o g e n e o u s s t r u c t u r e or a prescribed d i s t r i b u t i o n o f properties. Various finite-element t e c h n i q u e s have been used t o s t u d y : m e t a l - f l o w [ 1 - - 3 ] , forces [ 4 ] , strain [5] and t e m p e r a t u r e [6] distributions in metalf o r m i n g problems, and have been reviewed elsewhere [ 7, 8]. T h e p r e s e n t p a p e r a d o p t s an i n c r e m e n t a l elastic-plastic a p p r o a c h [ 9 , 1 0 ] in w h i c h the influence o f interface friction is i n c l u d e d b y the a d d i t i o n o f a n a r r o w layer o f elements on a n y surface o f t h e m o d e l w h e r e friction is p r e s e n t (see a p p e n d i x ) . The stiffnesses o f these layers are m o d i f i e d b y a f u n c t i o n o f the interface shear ratio, m. This t e c h n i q u e was s h o w n t o be valid f o r t h e analysis of the c o m p r e s s i o n o f rings in the ring-test [ 10], b u t until n o w has n o t been used f o r a n y o t h e r g e o m e t r y .
146 PUNCH
CO,,~AINEP
e
CONTAINER8
PJNCH
~:
Im~
/ ~ / / ~ / / ~:'BILLET
Fig. 1. Initial position (a) and deformed state (b) in axisymmetric rim-disc forging. Fig. 2. Finite-element mesh, including extra layer of 'friction' elements, representing one half of a meridian plane.
The rim-disc forging configuration is shown in Fig. 1 and the finite-element representation illustrated in Fig. 2. Stress-strain curve for aluminium Commercially-pure aluminium billets were machined to various height/ diameter ratios in their as-received condition. A Cook-and-Larke procedure [ 11] was used to evaluate the stress-strain curve illustrated in Fig. 3. An empirical relationship that was found to fit the experimental curve reasonably well is: = oy + 21.62 ln(~/0.00624)
(1)
where ~ and ~ are the current generalised stress and generalised strain, respectively, and ay is the initial yield stress. 200 STRESS 100 : N/mm 2
0
1
2 3 4 5 NATURAL STRAIN
6
7
8
Fig. 3. Experimental stress-strain results (A~) and representation ( - - ) used as input to finite-element program.
147
Experimental tim-disc forging The aluminium specimens, 39 m m dia. X 13.3 mm high, were p r o d u c e d in two halves with a rectangular grid etched o n t o the meridian plane of one half [ 12]. The two halves of the specimen were then relocated in the closely fitting container. Lanolin was applied to all external surfaces of the billet to provide good lubrication. The specimens were then d e f o r m e d to the required level at a very low strain-rate. After removal of the specimens from the container, the d e f o r m a t i o n pattern could be determined from observation of the d e f o r m e d grid (Fig. 4).
Fig. 4. E x p e r i m e n t a l d e f o r m a t i o n of initially square grid; (a) 10%, (b) 19%, (c) 26%: The grid lines shown are selected from an original experimental grid consisting of 38 vertical and 14 horizontal lines.
The interface shear factor m, for the lanolin lubrication used in these experiments, was determined from a ring-test using a ring-material of commercially-pure aluminium with lanolin as the lubricant [ 1]; the m-factor was f o u n d to be 0.04. The experimental hardness distributions expressed in terms of the Vickers Pyramid N u m b e r were obtained with a 5 kg load on a manual macro-hardness testing machine. Subsequent to the post-process heat t r e a t m e n t (annealing at 450°C for 2 hours), the specimens were repolished on a 600 grit disc and after cleaning were immersed in a solution of cupric chloride (125 gm) and water (1000 gm). T h e y were then washed in water and immersed in a 50% nitric acid solution to remove any of the c oppe r deposit and reveal the grain structure of the forging. The specimens were finally cleaned with acetone and dried.
148
Discussion of results Figure 5 illustrates the very good correlation found between the experimental and theoretical forces. In each case a steadily increasing force is found although the theory underestimates the actual force at the 4--8% deformation (reduction in height) range. This may be due to the theoretical assumption that the lubricant is always present, represented by the initially specified interface shear factor m. It is extremely difficult to ascertain whether full lubrication was maintained in the experiments, although with this type of forging it is likely that it was not. 2OO
I
r
[
FORCE
100
I,',N
0
i 0
5
10 15 OEFORMAT[ON
I 20
25
30
%
Fig. 5. Forging force predicted by finite-element m e t h o d ( - - ) mined ( ........>).
and e x p e r i m e n t a l l y deter-
The theoretical predictions of internal metal-flow illustrated in Fig. 6 indicate the formation of a dead metal region beneath the punch. This type of deformation pattern is frequently observed in other forging configurations
'iiiiiiiiilli'~ trait
t tl
I 1.24-~---~t
J [ I I I I I I.I~-t--~['T~.L.IMI I I I I I I I ~ ~ _ L I [ l l l l l
I
. . . . . . . .
illjiiiiill
, , , , , , , , ,
...... Jlilli' ]l[Jll,,,,
I
Fig. 6. Finite-element predictions of internal deformation; (a) 10%, (b) 16%, (c) 21%, (d) 26%.
149
where a flat punch is used. Another region subject to little deformation is at the base, near to the right-hand corner of the billet. The majority of the deformation is therefore restricted to a diagonal region running from the base-centre to the punch corner. This t y p e of deformation is clearly shown on the experimental grids in Fig. 4. The only major discrepancy between theory and experiment is in the metal-flow around the sharp punch comer. The predictions clearly underestimate the high shearing occurring just below the punch c o m e r and consequently predict a much more curved free surface than is actually found. It is thought that this restriction is partly a result of using the triangular elements in the arrangement shown in Fig. 2. A change to higher-order elements may reduce this problem.
FIG 7a
F
G
FIG 8a
O
C
F
I
C
D
FIG 7b
E FIG 8b
E
A
0
E
A
B 0
Fig. 7. Finite-element-predicted generalised strain distributions (a) 10%, (b) 26%. Strain ranges, (A) 0.61--1.03; (B) 0.46--0.60; (C) 0.31--0.45; (D) 0.16--0.30; (E) 0.11--0.15; (F) 0.06--0.10; (G) 0.04--0.05. Fig. 8. Finite-element-predicted generalised stress distributions (a) 10%, (b) 26%. Stress ranges, (A) 156--165 N/mm2; (B) 146--155; (C) 136--145; (D) 126--135; (E) 116--125; (F) 101--115; (G) 94--100.
150
This pattern of internal deformation is obviously reflected in the strain and stress distributions, shown in Figs. 7 and 8, respectively. The highest strains appear around the punch corner as it penetrates the billet and the lowest strains in the region beneath the punch and close to the lower righthand corner. The most significant results are the hardness distributions, shown in Figs. 9 and 10. These hardness values are of particular importance in assessing the
~ 30(32) 3003)
30(31) 29(32) 30(29) 40(3~ 39(36)
0(30)
37(33) 33(34) 34(34)
40(39) 35(36)
36(34) 3£(39) 36(37) 35(34) 3404)
30(30) 3002)
31(34) 32(32)
31(33) 3203)
33(35) 36(36) 37(35)
35(35) 36(35) 34(35) 33(35)
33(37) 3404)
35(34) 36(34) 32(34)
33(32) 32(32) 32(33) 32(36)
3101)
31(32)
37(40) 36(36) 35(37) 33(32) 31(32.3 29(30) 28(30) 3001)
30(33
36(39
37(37) 36(36) 34(35)
33(34)
31(31) 30(31)
Fig. 9. Comparison of finite-element-predicted hardness and experimental hardness (in parentheses) distributions (VPN) at 10% deformation.
I 37(30) 36(28) 36(29)
36(31)
40(43)
36(32) 49(45) 45(44) 40(39) 46(41) 44(43)
40(34)
41(45)
37(31) 37(30) 37(32) 39(36) 44(33) 44(41) 42(41) 40(40) 40(37)
38(35) 4O04)
4O04)
43(40
44(37) 40(37) 40(39)39(39)
39(39)
42(39) 42(38) 4307)
42(38) 40(39) 38(39) 37(34) 38(38) 38(41)
44(40)
40(39) 38(38) 3604)
44 (41) 42 (40)
3504)
36(35) 38(40)
Fig. 10. Comparison of finite-element-predicted hardness and experimental (in parentheses) distributions (VPN) at 26% deformation.
151 suitability of any c o m p o n e n t for a particular application. The theoretical hardness values were evaluated from the well known relationship [ 13], VPN = 2.967 ~
kgf/mm 2
(2)
At a reduction of height of 10% (Fig. 9), the correlation between the finiteelement prediction and the experimental distribution of the hardness values is very good, with an average variation of 3.8%. When the height reduction is increased to 26% the correlation -- although good -- is not as close, with an average variation of 8.6%. The predicted values generally appear to be higher than the experimental, especially in the highly deformed region. However, the experimental hardness values show surprisingly little change as the height reduction is increased from 10% to 26%. This may be due in part to a reduction in the strain-hardening of the aluminium due to temperature increases in the highly worked region, which the current finite-element program does not allow for. The restriction to flow around the punch corner is also likely to contribute to the high strains in the theoretical model. Despite this minor discrepancy, the actual deformation patterns and hardness distributions correlate very closely in both cases. The grain structure on the meridian plane of the specimen illustrated in Fig. 11 clearly shows the result of inhomogeneous deformation, indicated both by the experimentally deformed grids and the finite-element predictions. The corresponding predicted strain distribution is shown in Fig. 7 (b). The critical strain for large grain growth in aluminium is between 5 and 10%, so that the areas of lowest strain should therefore contain the largest grains and the band of intense shear the smallest. This arrangement is clearly revealed on the etched specimen in Fig. 11. It is very interesting to observe a closely similar pattern in a recrystallised aiuminium alloy extrusion produced under industrial conditions (Fig. 12).
Fig. 11. Etched specimen showing grain structure and approximate limits of regions with little deformation.
152
Fig. 12. Industrially produced aluminium extrusion showing regions of little deformation at the extrusion base and in the extreme sections of the annulus (courtesy of High Duty Alloys Forgings Ltd.). Conclusions T h e forging o f a disc with an annular c i r c u m f e r e n t i a l rim has b e e n examined in detail. An elastic-plastic f i n i t e - e l e m e n t m e t h o d has been used t o p r e d i c t the p u n c h forces, metal-flow, strain, stress and hardness distributions and to indicate t h e zones o f large grain growth. Close c o r r e l a t i o n was f o u n d with the e x p e r i m e n t s using c o m m e r c i a l l y p u r e a l u m i n i u m as a m o d e l material. T h e i n t r o d u c t i o n o f surface e l e m e n t layers with a stiffness m o d i f i c a t i o n f a c t o r based o n the interface shear ratio appears t o be quite successful. Rim-disc forging is typical o f processes displaying i n h o m o g e n e o u s deformation, and shows a band o f intense shear f r o m the base-centre to the p u n c h
153 c o m e r . T h e finite e l e m e n t p r e d i c t i o n s clearly indicate this t y p e o f i n h o m o geneous structure and predict with good accuracy the distribution of the p r o p e r t i e s o f the forging. A c c u r a t e d e t e r m i n a t i o n o f t h e strain a n d stress d i s t r i b u t i o n t h r o u g h o u t t h e forging c o u l d lead t o f u r t h e r d e v e l o p m e n t s o f this t y p e o f n u m e r i c a l analysis. Particularly o f i n t e r e s t is t h e p o t e n t i a l o f this a p p r o a c h f o r predicting f l o w and p o s s i b l y f r a c t u r e in m e t a l f o r m i n g processes, especially in cases w h e r e a s e c o n d o p e r a t i o n o f a d i f f e r e n t n a t u r e is p e r f o r m e d o n t h e first-stage forging.
Acknowledgements G r a t i t u d e is e x p r e s s e d t o t h e Science R e s e a r c h Council f o r f i n a n c i n g the r e s e a r c h a n d t o Mr A. D o w n i n g f o r assistance with t h e e x p e r i m e n t a l w o r k . T h a n k s are also d u e to P r o f e s s o r J.M. A l e x a n d e r f o r his v a l u a b l e c o m m e n t s o n t h e h a r d n e s s p r e d i c t i o n p r o c e d u r e , a n d t o High D u t y A l l o y s Ltd. f o r p e r m i s s i o n to r e p r o d u c e Fig. 12.
References 1 P. Hartley, C.E.N. Sturgess and G.W. Rowe, Proc. 20th Int. MTDR Conf., Birmingham (1979), p. 157, Macmillan, London (1980). 2 E.H. Lee, R.L. Mallett and W.H. Yang, Comp. Meth. App. Mech. Eng., 10 (1977) 339. 3 H. Matsumoto, S.I. Oh and S. Kobayashi, Proc. 18th Int. MTDR Conf., Imperial College, London (1977), p. 3, Macmillan, London (1978). 4 O.C. Zienkiewicz, P.C. Jain and E. Onate, Int. J. Solids Struct., 14 (1978) 15. 5 S.N. Shah and S. Kobayashi, Proc. 15th MTDR Conf., Birmingham (1974), p. 603, Macmillan, London (1975). 6 F.W. Sharman, Electricity Council Research Report R581 (1975). 7 J.M. Alexander and J.W.H. Price, Proc. 18th MTDR Conf., London (1977), p. 267, Macmillan, London {1978). 8 H. Takahashi and S. Kobayashi, Proc. 5th NAMRC Conf., Amherst, Massachusetts (1977), p. 87. 9 P. Hartley, Ph.D. Thesis, The University of Birmingham, U.K. (1979). 10 P. Hartley, C.E.N. Sturgess and G.W. Rowe, Int. J. Mech. Sci., 21 (1979) 301. 11 M. Cook and E.C. Larke, J. Inst. Met., 71 (1945) 371. 12 G.W. Rowe, I.M. Desai and H.S. Shin, Proc. 15th Int. MTDR Conf., Birmingham (1974), p. 417, Macmillan, London (1975). 13 F.P. Bowden and D. Tabor, The Friction and Lubrication of Solids, Oxford University Press, 1964. Appendix T h e f i n i t e - e l e m e n t m e s h illustrated in Fig. 1 o f the t e x t c o n s i s t e d o f 8 8 8 a x i - s y m m e t r i c t r i a n g u l a r e l e m e n t s w i t h 506 n o d a l points. This p r o g r a m req u i r e d 2 . 7 5 s e c o n d s o f c o m p u t e r t i m e o n a CDC 7 6 0 0 c o m p u t e r p e r 0.25% d e f o r m a t i o n i n c r e m e n t . An i n c r e m e n t a l t y p e o f elastic-plastic analysis was
154 adopted in which no iterations were involved. In this approach the incremental strain is related to the incremental nodal point displacements in the form {Ae} = [B]{AS}
(A1)
where {Ae} is the strain vector, {AS} is the displacement vector and [B] is the co-ordinate matrix formulated by assuming a linear-velocity displacement of the nodal points. The incremental stress is related to incremental strain in the form, (Ao} = [DI{Ae}
(A2)
where (Ao} is the stress vector and [D] is the elastic-plastic stress-strain matrix formulated from the Prandtl-Reuss equations for deviatoric strain and stress increments. Applying a virtual-work formulation will yield, {&Q}n = [g]n_l{AS}n
(A3)
where {AQ} is the vector of applied forces, n is the increment number and [K] is the stiffness matrix evaluated from, [g] = [B]T[D] [B]v
(A4)
where v is the element volume. The resulting large number of simultaneous equations are solved by a Gauss direct-elimination technique. The influence of friction is included in the model by the addition of a narrow layer of elements on any interface where friction is present. The stiffnesses of these layers were then modified by a function, 9, of the interface shear ratio, m, i.e. {3 = r n / ( 1 - m ) .
(A5)
This required the re-formulation of eqn. (A3) to, {AQ} n = ([K]Bn_ 1 +~[g]Ln_l)(AS}n
(A6)
where B and L indicate the billet and layer, respectively. More complete details of the finite-element and friction techniques used here can be found in Refs. 1 and 10.