Civil Engineering Reference
In-Depth Information
shown schematically in Fig.
5.1
. Friction is present at the workpiece/die interfaces,
thus causing the part to bulge into a specific barrel shape. This barreling effect is
neglected in the initial EAM modeling analysis.
The following major simplifications and assumptions are used throughout the
derivations in this study:
• The material is homogeneous and isotropic.
• The elastic deformation is negligible and the power law is used for the low
stress of the material.
• The von Mises yield criterion is applied.
• The friction of the workpiece/die interfaces follows Coulomb's friction law.
Sliding friction conditions exist, and barreling is ignored.
• The speciic heat/resistivity of the material is independent from temperature.
• The strain rate sensitivity is neglected since previous experimental investiga-
tions found that the temperature rise is not high enough for the process to be
considered hot forming.
• Volume is maintained.
• The lubrication regime is unaffected by the presence of electricity.
5.1.3 Effective Stress and Strain—Classical Compression
Test
The energy of a process is the total work needed to perform that process. The power
of a process relates the required work to a particular speed at which the work is
performed. The power needed to plastically deform a cylindrical billet is found by
employing the upper bound analysis method as presented by Avitzur [
7
]. The objec-
tive of this method is to determine the strain rate field that minimizes Eq. (
5.2
):
J
∗
=
W
i
+
W
s
+
W
b
+
W
k
+
W
p
+
W
γ
(5.2)
where
J
∗
is the upper bound on power,
W
i
is the internal power of deformation,
W
s
is the shear power given by the discontinuity of power to overcome opposing
external forces (back stress),
W
k
is the inertia velocity,
W
b
is the power due to the
inertia forces,
W
p
is the pore opening power, usually neglected under the volume
constancy assumption, and
W
γ
is the surface power that includes the rate of intro-
duction of new surfaces in the deformation process. This component can also be
assumed negligible in bulk deformation, but will be of more importance as this
method is extended to sheet metal forming. The power components were computed
by Avitzur [
7
] for the ideal case where no bulge is formed, and the shape of the bil-
let is assumed to remain cylindrical throughout the deformation process, unaffected
by the interface friction. Using the average pressure approximation, where the pres-
sure on the workpiece is assumed to be constant across the cross section, the follow-
ing equation for total power needed for compression can be derived as Eq. (
5.3
):
