Kinematic dependences (8) make it possible to calculate the distribution of the
radial velocity difference V
r
at the interface of the layers along the deformation zone,
the length of which is determined by the coordinate r. It is shown that:
- the maximum difference in the rates of movement is observed at the entrance of
metals to the deformation site;
- a layer that has a smaller yield boundary is ahead of a layer with a larger yield
boundary, due to the fact that it is thinner to a greater extent;
- at the output of the deformation cell, the velocity of the layers is aligned 10, 11-
15.
It is shown that taking into account the friction forces on the boundary surfaces of
the layers provides a more accurate solution. For this purpose, the analysis of the
energy-power parameters of drawing with refinement, taking into account the friction
forces on the boundary surfaces using the method of power balance.
The power balance equation for the deformation cell of the first layer has the form:
,
зс
R
зс
R
мс
тр
тр
іB
іH
і
r
і
rі
і
z
N
N
N
N
N
N
N
N
+
+
+
+
+
+
=
(11)
When solving the equation (11), the friction forces are taken into account by the
friction coefficient according to the Amonton-Coulomb law on the boundary surfaces
of the workpiece with the matrix –
3
, and a punch –
1
, and between the layers of the
blank –
2
. Then the total deformation power of both layers is determined by the
dependence:
(
)
(
)
(
)
(
)
.
ln
ln
1
ln
1
3
2
ln
ln
1
ln
1
3
2
1
2
1
1
1
0
0
1
2
2
1
0
0
1
2
3
0
2
1
1
1
1
1
0
0
1
1
1
1
0
0
1
1
2
0
1
2
1
−
+
+
−
−
−
−
−
−
−
−
−
+
+
+
+
−
−
−
−
−
=
B
k
H
H
B
B
k
k
H
B
k
H
B
k
k
H
B
k
H
S
B
H
H
B
B
k
H
B
H
B
k
H
B
H
S
S
S
S
S
S
S
S
ctg
S
S
S
S
S
S
S
ctg
S
S
S
S
S
V
S
S
S
S
S
ctg
S
S
S
S
S
ctg
S
S
S
S
V
N
z
(12)
Equation (12) makes it possible, taking into account the extreme energy principles
of plastic deformation, to determine the unknown thickness of the first layer S
1B
after
drawing. Its determination is made by minimizing the power of the process:
.
0
1
=
B
S
N
z
(13)
Equation (13) is solved by a numerical method of simple iterations. The
comparison of calculation results of the layer thickness with a lower limit of fluidity at
a stretching with thinning of the workpiece with the initial (Ѕ
0Н
=2.8 mm Ѕ
1Н
= Ѕ
2Н
=1.4
mm) and end S
k
=1.12 mm thickness of layers (Fig. 2) shows that when taking into
account the forces of friction thinning is manifested to a greater extent than without
taking into account the forces of friction. The calculation error does not exceed
10...11%.
- 523 -