Civil Engineering Reference
In-Depth Information
1
+
2
µ
r
0
3
h
inst
J
∗
= π
r
inst
σ
u
=
F u
(5.3)
where
r
inst
and
h
inst
are the instantaneous radius and height of the workpiece,
σ
is the
effective flow stress of the material,
u
is the velocity of the compressive die,
r
0
is the
initial radius of the billet, and
F
is the forming load. When the applied load and die
velocity are known, the pressure exerted by the die is determined as Eq. (
5.4
):
F
π
r
inst
1
+
2
µ
r
0
3
h
inst
p
=
=σ
(5.4)
The power law will be used to formulate the relation between the effective stress
and strain, thus the pressure will be given by Eq. (
5.5
):
1
+
2
µ
r
0
3
h
inst
p
=
C
ε
n
(5.5)
By employing the slab analysis method and simple balance of forces in the axial
direction, along with the von Mises criterion, the compressive and effective
stresses are determined from Eq. (
5.6
):
σ
z
=
p
,
σ
θ
= σ
r
=
p
−σ
,
ε
z
=−
2
ε
θ
=−
2
ε
r
,
(5.6)
h
0
h
inst
ε =−ε
z
=
ln
,
where
σ
z
,
σ
θ
,
σ
r
and
ε
z
,
ε
θ
,
ε
r
are the principal stresses and strains in axial, hoop,
and radial direction, respectively, and
h
0
is the initial height of the billet.
5.1.4 Effective Stress and Strain—Electrically Assisted
Compression Test
The flow stress required to deform a specimen to a desired strain is lowered by
applying electric current during the process. In this section, the effective electrical
energy utilized in reducing this stress is formulated.
When electricity is applied during deformation, a part of the electrical energy
is imparted into the mechanical deformation process. Thus, the total power con-
sumed by the process will be given by Eq. (
5.7
):
J
∗
=
J
m
+
J
e
(5.7)
where
J
m
is the mechanical component given by the product of the forming load and
the die velocity, and
J
e
is the effective (usable) electrical power, as shown in Eq. (
5.8
):
J
e
=
ξ
·
P
e
(5.8)
