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the plastic yield
dA
p
/dt
and by the yield stress
σ
Y
as
λ
=
3
2
σ
2
Y
dA
p
dt
.
A simplified method of Prandtl – Reiss [6] incremental plasticity
equations can be useful while applying an approach of reducing the vector
of the stress tensor deviator to the yield circle for solving elastoplastic
medium mechanical problems. In accordance with this approach, the
components of the stress tensor deviator were calculated assuming the
elastic behavior of the material. The following set of equations was used:
ds
z
dt
= 2
G
˙
ε
z
+
1
3
ρ
dρ
dt
+
δ
z
;
ds
r
dt
= 2
G
˙
ε
r
+
1
3
ρ
dρ
dt
+
δ
r
;
ds
θ
dt
= 2
G
˙
ε
θ
+
1
3
ρ
dρ
dt
;
dτ
rz
dt
=
G
˙
ε
rz
+
δ
rz
,
here
δ
z
,
δ
z
,
δ
rz
are corrections of the stress tensor deviator components
connected to the rotation of a specified medium element as a rigid unit and
calculated according to the following equations:
δ
z
=
τ
rz
∂v
z
∂r
−
∂v
r
∂z
;
δ
r
=
τ
rz
∂v
r
∂z
−
∂v
z
∂r
;
δ
rz
=
1
2
(
s
z
−
s
r
)
∂v
r
∂z
−
∂v
z
∂r
.
Then both the Mises yield is checked. The stress tensor deviator
components are corrected if required. An auxiliary function is calculated
J
= 2
s
2
r
+
s
2
z
+
s
2
θ
+ 2
τ
2
rz
.
In case the values of
s
z
,
s
r
,
s
θ
,
τ
rz
, which were calculated assuming
material elastic behavior, conform to the condition
J >
(2
/
3)
σY
2
(it
corresponds to the plastic flow of the material), they are multiplied by
p
2
/
3(3
J
)
σ
Y
[6] for correction.
With the use of the values
s
z
,
s
r
,
s
θ
, calculated according to the
incremental plasticity and pressure
p
, normal components
σ
z
=
s
z
−
p
,
σ
r
=
s
r
−
p
,
σ
θ
=
s
θ
−
p
of the stress tensor are calculated, these components
are included in the equations of axial motion of the material particles.
ISSN 0236-3941. HERALD of the BMSTU Series “Mechanical Engineering”. 2015. No. 1 67