Civil Engineering Reference
In-Depth Information
FIGURE 5.16
The forging problem.
where:
v is the velocity field
is the strain rate tensor,
˙
˙
1/2 (grad(
v
)
grad
t
(
v
))
˙
˙
is the equivalent strain rate:
˙
:
(5.15)
()
˙
2
3
is the Cauchy stress tensor
s
is the deviatoric part of
,
s
I
and
is the hydrostatic pressure,
1/3 trace(
)
K
is the consistency of the material
m
is the strain rate sensitivity coefficient
At the interface between the tools and the part,
∂
c
, as the contact pressure is often high, the friction is
assumed to follow a Norton law:
q
1
K
v
t
v
t
(5.16)
where:
is the shear stress:
v
t
is the relative tangential velocity:
v
t
[
(
v
die
)
.
t
]
t
(5.17)
v
die
is the prescribed velocity of the die
and
q
are the friction law coefficients
At any time in the process, the forged workpiece
is in equilibrium between the forging dies, the
∂
c
part of the surface, while
forging conditions, inertia effects and gravity forces are negligible, so the balance equations are:
∂
f
div
()
0on
(5.18)
q
1
(
VV
die
)
n
0
and
K
v
t
v
t
,
on
c
(5.19)
n
0
on
f
(5.20)
Variational Formulation and Finite Element Discretization
In order to solve Eqs. (5.18) and (5.12), they are written again in a variational form, which is almost
equivalent to the virtual work principle. In the standard way for incompressible flow problems, a veloc-
ity/pressure formulation is used. The set of kinetically admissible velocity fields (which satisfy the contact
