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On unsymmetrical extrusion in plane strain
Journal of tile Mechanics and Physics of Solids, 1955, Vol. 3, pp. 189 to 196.
ON
UNSYMMETRICAL
EXTRUSION
Pergamon Press Ltd., London.
IN
PLANE
STRAIN
By A. P. GREEN" Tube Investments Research Laboratories, Hinxton Hall, Cambridge (Received 2nd February, 1955)
SUMMARY
SLIP-LINEfields are proposed and extrusion pressures are calculated for unsymmetrical extrusion through a square die, situated either at the end or in the side of a container with smooth or rough walls. A plastic-rigid non-hardening material is assumed. Plasticine experiments are described which reveal the deformation and confirm the theory.
1.
INTRODUCTION
PLANE strain solutions* have been proposed by HILL (1948) for extrusion through a square die placed symmetrically at the end of a straight parallel-sided container which has either smooth or rough walls. A non-hardening plastic-rigid material is assumed. The state of stress in a plane of flow is pure shear combined with a hydrostatic pressure, p. In the plastic regions the m a x i m u m shear stress is a constant, k, but in general p varies. I n this paper we discuss how these solutions m u s t be modified if the aperture of the die is not central with respect to the container. Calculations of extrusion pressures include those for symmetrical extrusion from a rough container, which were not calculated b y HILL. Solutions are also proposed for extrusion through an aperture in the side instead of a t the end of the container. I t often happens in extrusion t h a t regions o f " dead " material are left behind in the corners of the container as a steady state of flow is initiated (see for example Fig. 2), and thereafter constrain the flow as if they formed a rough die. As HILL (1948) has remarked, the shapes of these regions depend on so m a n y variables, such as friction over the die, t e m p e r a t u r e and speed of extrusion, etc., t h a t it is impossible to predict t h e m theoretically. Therefore, we assume boundaries for t h e dead regions which satisfy frictional conditions only where they meet the container walls, and are reasonably close to what is actually observed, b u t are otherwise chosen for simplicity. They are also assumed, in the solutions presented here, to be continuous slip lines. The stresses are not examined in the rigid regions, including the dead regions, and to this extent the solutions are theoretically incomplete. In spite of the simplifying assumptions and the incompleteness of the theory, the deformation predicted corresponds closely to t h a t observed in various model experiments with plasticine, which deforms in plane strain like a metal (Ga~.EN #For an account of the theory of plane plastic strain, see R. HILL, The Mathematical Theory of Plasticity, Ch. VI (Clarendon Press 1950).
189
190
A.P.
GREEN
1951). The deformation of the plasticine is revealed b y a square grid (1/10 or 1/5 inch squares) stamped in ink on a central plane of flow before extrusion.
2.
END EXTRUSION
The widths of the container and die aperture are denoted b y D and 2d respectively, and the widths of the die on either side of the aperture b y b and c. The fractional reduction, r, is ( D - - 2 d ) / D and the eccentricity of the die is conveniently defined as ~ : (b - - c)/(b ~- c), ranging from zero (symmetrical) to a m a x i m u m of one.
EXTRUDED PRODUCT
5' /
E~ J
DEAD REGIONS c%
b
C
Fig. 1. Slip-line fields for end-extrusion with rough container walls : (a) HILL'S symmetrical field; (b) an unsymmetrical field and (e) its hodograph.
(a) R o u g h container walls. The term ' rough ' is taken to mean that, under the high pressure involved, the frictional stress reaches the value k wherever the billet moves relative to the container walls. Hence, the slip lines of one family meet the walls tangentially. A simple modification of HILL'S symmetrical slip-line field (Fig. la) is shown in Fig. l b (slip lines across which the velocity is discontinuous are drawn thicker in this as in other Figures). The form of the field is unaltered, being defined b y the equal circular arcs B E z A 1 and B E 2 - 4 2 but it now extends farther on one side than on the other, with corresponding unequal dead regions in the corners. Equilibrium of the extruded material demands t h a t p ~ k on the straight slip lines B O 1 and B O 2. The stress at a n y other point in the field is deduced by means of the well known H e n c k y relations along the slip lines. The hodograph of the velocity field (GREEN 1954, PRAGER 1953) is shown in Fig. lc, from which it is clear t h a t the b o u n d a r y conditions are satisfied and the material is extruded straight but obliquely towards the side to which the aperture is closer.
On unsymmetrical extrusion in plane strain
191
HILL'S symmetrical solution has already been closely confirmed, including the assumed shape of the dead region, by a plasticine experiment (GREEN 1951), The present unsymmetrical solution is not so precisely compared with experiment, but Fig. 2 shows that the extent of the plastic regions and the shapes of the dead regions in the plasticine specimens correspond qualitatively to the theory. The pressures, P, on the extrusion ram and the depths of the dead regions, C X = l, have been calculated between r = 0 and r = 0.78 for the symmetrical case by integrating the components of force parallel to the container walls acting across the slip lines OAC. These pressures correspond to positions of the ram R R coinciding with the ends of the dead regions CC ; if the ram is further back 0.6
0'5 ROUGH WALLS SMOOTH WALLS
. . . .
.¢/
7
o/
0.4
I
/
0
0"3
/ //
0.2
t
/
/ J
/
0"1 i 0
0.2
0.4.
0'6
0-8
~.0
Fig. 3. Increase in extrusion pressure due to eccentricity, versus eccentricity. an additional term 2k × R C / D must be added, representing the friction between the wall and the remaining length of billet, RC. The calculated values of P / 2 k and lid are given in the Table. The following empirical equation gives P / 2 k to within 2½% over the range of r considered : - -
P / 2 k = 1/%/2 + 1.83 In i-------r "
(1)
The pressure in eccentric extrusion can be deduced from equation (1) without further integration. The slip lines OA1C 1 and 0A2C2, with depths of dead regions 11 and 12 correspond respectively in symmetrical extrusion to reductions rl - -
r (1 + ~)
1 -q- ~r '
and
r2 _
r(1
-
~),
1 - - Er
(2)
and to extrusion pressures P1 and P2, say. Adding the forees across both these slip lines, the resultant pressure is found to be P~ =
½ [P, + P2 +
Er (P1 -- P2]"
(8)
To this must be added the pressure k ( 2 R X -- 11 -- 12)/D due to the frictional drag between the billet and the walls, the values of ll/d and 12/d being interpolated from the Table. Hence, using equations (1) and (2) to express P! and P2 in terms of r and c, the amount by which the total pressure in un~ymnletrieal extrus{on
192
A . P . GRrr~
exceeds t h a t in symmetrical extrusion for the same reduction and position of ram is A P = 1,88k [(1 + ~r) In (1 + ~r) + (1 -- ~r) In (1 -- ~r)] + k (1 -- r) (21 -- 11 -- ls)/d
/ I
(4)
The plots of A P / k against c in Fig. 8, for r ----0.2, 0.4, ant1 0-6, show t h a t it increases with increasing reduction and eccentricity. It is unaffected by the position of the ram until the ram approaches close to, or reaches, a dead region, when the slip line f i e l d will depend on frictional conditions on the ram. The problem is then no longer one of steady motion and its solution has not been attempted. I EXTRUDED PRODUCT
DEAD REGIONS a
]s
Fig. 4. (a) Slip-line field, and (b) its hodograph for unsymmetrical end-extrusion with smooth container walls in range (i). (b) .Smooth conlainer walls. Two types of solution for symmetrical ex
+r(1--r)
+ l - 8 8 1 n ( 1 - - ~ r ).
(5)
In unsymmetrical extrusion three ranges m u s t be considered i.e. (i) b and e > d, (ii) b and e < .d, Off) b > d, c < d.
ON
UNSYMMETRICAL
EXTRUSION
Pergamon Press Ltd., London.
IN
PLANE
STRAIN
By A. P. GREEN" Tube Investments Research Laboratories, Hinxton Hall, Cambridge (Received 2nd February, 1955)
SUMMARY
SLIP-LINEfields are proposed and extrusion pressures are calculated for unsymmetrical extrusion through a square die, situated either at the end or in the side of a container with smooth or rough walls. A plastic-rigid non-hardening material is assumed. Plasticine experiments are described which reveal the deformation and confirm the theory.
1.
INTRODUCTION
PLANE strain solutions* have been proposed by HILL (1948) for extrusion through a square die placed symmetrically at the end of a straight parallel-sided container which has either smooth or rough walls. A non-hardening plastic-rigid material is assumed. The state of stress in a plane of flow is pure shear combined with a hydrostatic pressure, p. In the plastic regions the m a x i m u m shear stress is a constant, k, but in general p varies. I n this paper we discuss how these solutions m u s t be modified if the aperture of the die is not central with respect to the container. Calculations of extrusion pressures include those for symmetrical extrusion from a rough container, which were not calculated b y HILL. Solutions are also proposed for extrusion through an aperture in the side instead of a t the end of the container. I t often happens in extrusion t h a t regions o f " dead " material are left behind in the corners of the container as a steady state of flow is initiated (see for example Fig. 2), and thereafter constrain the flow as if they formed a rough die. As HILL (1948) has remarked, the shapes of these regions depend on so m a n y variables, such as friction over the die, t e m p e r a t u r e and speed of extrusion, etc., t h a t it is impossible to predict t h e m theoretically. Therefore, we assume boundaries for t h e dead regions which satisfy frictional conditions only where they meet the container walls, and are reasonably close to what is actually observed, b u t are otherwise chosen for simplicity. They are also assumed, in the solutions presented here, to be continuous slip lines. The stresses are not examined in the rigid regions, including the dead regions, and to this extent the solutions are theoretically incomplete. In spite of the simplifying assumptions and the incompleteness of the theory, the deformation predicted corresponds closely to t h a t observed in various model experiments with plasticine, which deforms in plane strain like a metal (Ga~.EN #For an account of the theory of plane plastic strain, see R. HILL, The Mathematical Theory of Plasticity, Ch. VI (Clarendon Press 1950).
189
190
A.P.
GREEN
1951). The deformation of the plasticine is revealed b y a square grid (1/10 or 1/5 inch squares) stamped in ink on a central plane of flow before extrusion.
2.
END EXTRUSION
The widths of the container and die aperture are denoted b y D and 2d respectively, and the widths of the die on either side of the aperture b y b and c. The fractional reduction, r, is ( D - - 2 d ) / D and the eccentricity of the die is conveniently defined as ~ : (b - - c)/(b ~- c), ranging from zero (symmetrical) to a m a x i m u m of one.
EXTRUDED PRODUCT
5' /
E~ J
DEAD REGIONS c%
b
C
Fig. 1. Slip-line fields for end-extrusion with rough container walls : (a) HILL'S symmetrical field; (b) an unsymmetrical field and (e) its hodograph.
(a) R o u g h container walls. The term ' rough ' is taken to mean that, under the high pressure involved, the frictional stress reaches the value k wherever the billet moves relative to the container walls. Hence, the slip lines of one family meet the walls tangentially. A simple modification of HILL'S symmetrical slip-line field (Fig. la) is shown in Fig. l b (slip lines across which the velocity is discontinuous are drawn thicker in this as in other Figures). The form of the field is unaltered, being defined b y the equal circular arcs B E z A 1 and B E 2 - 4 2 but it now extends farther on one side than on the other, with corresponding unequal dead regions in the corners. Equilibrium of the extruded material demands t h a t p ~ k on the straight slip lines B O 1 and B O 2. The stress at a n y other point in the field is deduced by means of the well known H e n c k y relations along the slip lines. The hodograph of the velocity field (GREEN 1954, PRAGER 1953) is shown in Fig. lc, from which it is clear t h a t the b o u n d a r y conditions are satisfied and the material is extruded straight but obliquely towards the side to which the aperture is closer.
On unsymmetrical extrusion in plane strain
191
HILL'S symmetrical solution has already been closely confirmed, including the assumed shape of the dead region, by a plasticine experiment (GREEN 1951), The present unsymmetrical solution is not so precisely compared with experiment, but Fig. 2 shows that the extent of the plastic regions and the shapes of the dead regions in the plasticine specimens correspond qualitatively to the theory. The pressures, P, on the extrusion ram and the depths of the dead regions, C X = l, have been calculated between r = 0 and r = 0.78 for the symmetrical case by integrating the components of force parallel to the container walls acting across the slip lines OAC. These pressures correspond to positions of the ram R R coinciding with the ends of the dead regions CC ; if the ram is further back 0.6
0'5 ROUGH WALLS SMOOTH WALLS
. . . .
.¢/
7
o/
0.4
I
/
0
0"3
/ //
0.2
t
/
/ J
/
0"1 i 0
0.2
0.4.
0'6
0-8
~.0
Fig. 3. Increase in extrusion pressure due to eccentricity, versus eccentricity. an additional term 2k × R C / D must be added, representing the friction between the wall and the remaining length of billet, RC. The calculated values of P / 2 k and lid are given in the Table. The following empirical equation gives P / 2 k to within 2½% over the range of r considered : - -
P / 2 k = 1/%/2 + 1.83 In i-------r "
(1)
The pressure in eccentric extrusion can be deduced from equation (1) without further integration. The slip lines OA1C 1 and 0A2C2, with depths of dead regions 11 and 12 correspond respectively in symmetrical extrusion to reductions rl - -
r (1 + ~)
1 -q- ~r '
and
r2 _
r(1
-
~),
1 - - Er
(2)
and to extrusion pressures P1 and P2, say. Adding the forees across both these slip lines, the resultant pressure is found to be P~ =
½ [P, + P2 +
Er (P1 -- P2]"
(8)
To this must be added the pressure k ( 2 R X -- 11 -- 12)/D due to the frictional drag between the billet and the walls, the values of ll/d and 12/d being interpolated from the Table. Hence, using equations (1) and (2) to express P! and P2 in terms of r and c, the amount by which the total pressure in un~ymnletrieal extrus{on
192
A . P . GRrr~
exceeds t h a t in symmetrical extrusion for the same reduction and position of ram is A P = 1,88k [(1 + ~r) In (1 + ~r) + (1 -- ~r) In (1 -- ~r)] + k (1 -- r) (21 -- 11 -- ls)/d
/ I
(4)
The plots of A P / k against c in Fig. 8, for r ----0.2, 0.4, ant1 0-6, show t h a t it increases with increasing reduction and eccentricity. It is unaffected by the position of the ram until the ram approaches close to, or reaches, a dead region, when the slip line f i e l d will depend on frictional conditions on the ram. The problem is then no longer one of steady motion and its solution has not been attempted. I EXTRUDED PRODUCT
DEAD REGIONS a
]s
Fig. 4. (a) Slip-line field, and (b) its hodograph for unsymmetrical end-extrusion with smooth container walls in range (i). (b) .Smooth conlainer walls. Two types of solution for symmetrical ex
+r(1--r)
+ l - 8 8 1 n ( 1 - - ~ r ).
(5)
In unsymmetrical extrusion three ranges m u s t be considered i.e. (i) b and e > d, (ii) b and e < .d, Off) b > d, c < d.