To determine the components of the stress tensor, the equations of connection
between the stress deviators are used D
and strain rates D
:
D
D
i
і
3
2
=
(4)
where
і
and
і
– accordingly, the intensity of strain rates and stresses.
We believe that the workpiece metal is an ideal plastic that the process is
performed under hot (isothermal) plastic deformation
s
i
=
Medium voltage defined from the equation proposed by M. Alekseev:
−
=
T
div
grad
і
і
3
2
(5)
where Т
– the tensor of velocities of deformations.
Taking into account the limiting condition, at r = R
Н
,
0
rr
=
, the stress tensor
components have the form:
;
ln
3
2
r
R
H
S
rr
=
;
ln
1
3
2
−
−
=
r
R
H
S
.
3
ln
3
2
S
H
S
zz
r
R
−
=
(6)
The stress distribution (6)corresponds to the dependencies that are obtained E.O.
Popov 7 drawing with thinning of monometal. This allowed to confirm accepted
hypotheses and assumptions.
The obtained results are used for theoretical analysis of two-layer metal pulling.
For this purpose, the following additional assumptions are made:
- with a decrease in the total wall thickness of a two-layer blank, depending on
the mechanical heterogeneity of the properties of its layers, there is an uneven decrease
in the thickness of each layer;
- elongation of the individual layers along the direction of movement of the punch
will not be the same. This leads to the mutual movement of the material particles of the
boundary surface of the layers relative to each other;
- to create a diffusion connection, it is necessary that at the end of the compression
zone of the deformation zone at r=R
В
the speed of mutual displacement of material
particles at top surface leveled.
The kinematic feature of the deformation of a two-layer metal is that at the exit of
the deformation site at r=R
В
due to the different mechanical properties of the layers,
the ratio of the thickness of the metal changes. This leads to the fact that the limiting
surface in the deformation zone is rotated by a certain angle relative to the radial
direction (Fig. 1). As a result, the calculated radii of the layers are changed.
When calculating the thickness of the layers after stretching, the physical
condition of the joint deformation of the layers in the deformation cell is taken into
account-the equality of the tangential stresses normal to the limit surface
ІI
I
=
.
- 521 -